Combinatorial optimization in geometry

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Advances in Applied Mathematics 31 (2003) 242–271

www.elsevier.com/locate/yaam

Combinatorial optimization in geometry

Igor Rivin

Mathematics Department, University of Manchester and Mathematics Department,Temple University, Philadelphia, PA 19122–6094, USA

Received 12 March 2002; accepted 18 April 2002

Abstract

In this paper we extend and unify the results of [Rivin, Ann. of Math. 143 (1996)] and [RAnn. of Math. 139 (1994)]. As a consequence, the results of [Rivin, Ann. of Math. 143 (1996generalized from the framework of ideal polyhedra inH3 to that of singular Euclidean structuressurfaces, possibly with an infinite number of singularities (by contrast, the results of [Rivin, Ann. oMath. 143 (1996)] can be viewed as applying to the case of non-singular structures on the disa finite number of distinguished points). This leads to a fairly complete understanding of the mspace of such Euclidean structures and thus, by the results of [Penner, Comm. Math. Phys. 11299–339; Epstein, Penner, J. Differential Geom. 27 (1988) 67–80; Näätänen, Penner, Bull. LMath. Soc. 6 (1991) 568–574] the author [Rivin, Ann. of Math. 139 (1994); Rivin, in: Lecture Nin Pure and Appl. Math., Vol. 156, 1994], and others, further insights into the geometry and topof the Riemann moduli space.

The basic objects studied are the canonicalDelaunaytriangulations associated to the aforemtioned Euclidean structures.

The basic tools, in addition to the results of [Rivin, Ann. of Math. 139 (1994)] and combinatorigeometry are methods of combinatorial optimization—linear programming and networkanalysis; hence the results mentioned above are not onlyeffectivebut alsoefficient. Some applicationsof these methods to three-dimensional topology are also given (to prove a result of Casson’s) 2002 Elsevier Inc. All rights reserved.

Keywords:Linear programming; Network flow; Moduli space; Euclidean structures; Hyperbolic structures;Delaunay triangulations

This is a slightly modified version of a preprint first distributed in September 1996.E-mail address:[email protected].

0196-8858/02/$ – see front matter 2002 Elsevier Inc. All rights reserved.doi:10.1016/S0196-8858(03)00093-9

I. Rivin / Advances in Applied Mathematics 31 (2003) 242–271 243

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Introduction

In the paper [21] we gave the following description of the angles of ideal polyhedH3: let P be a combinatorial polyhedron, and letA :E(P) → [0,π) be a function. Thenthere exists an ideal polyhedron combinatorially equivalent toP , such that the exterioangle at every edgee is given byA(e) if and only if the sum ofA(e) over all edges adjacento a vertex ofP is equal to 2π, while the sum ofA(e) over anynon-trivial cutsetof edgesof P (that is, a collection of edges which separates the 1-skeleton ofP , but which arenot all adjacent to the same vertex) is strictly greater than 2π. Furthermore, it was showin [20] that the dihedral angles determine the ideal polyhedron up to congruence.

It was observed in [19,20] that this was a special case of the problem of charactethe Delaunay tessellationsof singular Euclidean surfaces—there is a canonical waassociate ideal polyhedra to Delaunay triangulations of a convex flat disk with cpolygonal boundary. The general situation is described in detail below, but one of the goaof this paper is to extend the characterization above to the completely general case osingular Euclidean surfaces with boundary. This is the content of Theorems 3.1, 4.9, 4.1

Consider a surfaceS, equipped with a Euclidean (or, more generally, a similarstructureE, possibly with cone singularities. Assumethat there is a discrete collectioP = p1, . . . , pn of distinguished points onS, and assume thatP contains the conepoints of S. There is a canonical tessellation attached to the triple(S,E,P )—the so-called Delaunay tessellation (see, e.g., [6,20]). The moduli spaceM of such triplesis then naturally decomposed into disjoint subsetsMT , corresponding to the differencombinatorial typesT of the Delaunay tessellation. This is a canonical decomposof M. In the paper [20] I studied the subsetsMT , and showed that thedihedral anglesof the Delaunay triangulation are natural coordinates (moduli) for MT , which induce onMT the structure of a convex polytope.

The aforementioned decomposition of moduli space then becomes a polyhedcomplex, the top-dimensional cells of which corresponds to Delaunay tessellationsare triangulations, while pairs of adjacent top-dimensional cells differ combinatorially ba diagonal flip. This decomposition is closely related to the well-known Harer com(see, e.g., [11]). As mentioned above, the top-dimensional cells of this compleidentified along some of their lower dimensional faces, while other lower-dimensionfaces correspond to degenerations of the Euclidean structures of(S,E,P ). It is then clearthat the polyhedral structure of the cellsMT is of considerable interest. However, in [2only an indirect description was given—MT was shown to be a convex polytope by virtof being an image of another convex polytope under a fairly complicated linear mapmethods of [21] come from hyperbolic geometry and are based on the study of diangles ofcompacthyperbolic polyhedra in [17], so do not easily generalize to the casgeneral singular Euclidean and similarity structures alluded to above. In the currentmethods of mathematical programming and the results of [20] are used to give a comgeneral extension of the result of [21] (described in the beginning of this IntroductioDelaunay triangulations of arbitrary singular surfaces (Theorems 3.1, 4.9, 4.11). Sthe arguments do not depend on the results of [21], we have a different, essecombinatorial, proof of a principal result (Theorem 0.1) of that paper (Theorem 4.11The other result of [21]—the characterization of finite-volume polyhedra—is, seem

244 I. Rivin / Advances in Applied Mathematics 31 (2003) 242–271

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not accessible by current methods. The above-mentioned result permit us to get gounderstanding of the boundary structure of the cellsMT , and, consequently, ofM itself.SinceMT fibers (in multiple ways) over the moduli space of finite area hyperbolstructures onS, some information is obtained about the latter moduli space.

In addition, the methods, combined with a geometric estimate, allow us to gdescription of dihedral angles of Delaunay tessellations of(S,E,P ), where(S,P ) is notnecessarily of finite topological type (Theorem 6.1). This stops well short of solvinmoduli problem, unlike in the finite case, but a conjectural picture seems fairly clear.

The methods are also brought to bear onto some questions in combinatorial geoand to provide efficient algorithms for solving the “inverse problem” of determining wa combinatorial complex, or a combinatorial complex equipped with dihedral anglecan be realized as the Delaunay tessellation of a singular Euclidean surface.

In addition, we use our methods to prove some observations of Casson [3] on itriangulated 3-manifolds. That subject is not so far removed from the geometsimilarity and Euclidean structures on surfaces. Indeed, the basic idea of [20] is tothe similarity and Euclidean structures by means of constructing a canonical hypepolyhedral complex as a “cone” over the surface being studied.

The plan of the paper is as follows: In Section 1 the relevant definitions and rof [20] are recalled. In Section 2 we describe a set of constraints which must be satisthe dihedral angles of any (not necessarily Delaunay) triangulation. In Section 3 wethat these constraints are actually sufficient under the assumption that the triangulDelaunay, and refine them to a minimal set of constraints. In Section 6 we show horesults apply to ideal polyhedra, and in particular to characterize infinite ideal polyhedra.Section 7 we comment on the boundary structure of the moduli space of singular Eucstructures, and describe a correspondence between the Euclidean and hyperbolic stwhich hopefully clarifies the picture. In Section 10 we apply the methods of Sectto the study of ideal triangulations of 3-manifolds. In Section 5 we give a networkinterpretation of the results of Section 4. In addition to the intrinsic interest, this aus to give efficient algorithms for deciding whether a weighed graph is the 1-skelea Delaunay triangulation (with weights being the dihedral angles). These computaissues are discussed in Section 8. In Section 9 we discuss some combinatorial-geapplications of the results of Section 4.

1. Background

1.1. Singular similarity structures

Consider an oriented surfaceS, possibly with boundary, and with a numberdistinguished pointsp1, . . . , pn. A similarity structureon S is given by an atlas foS\p1, . . . , pn, such that the transition maps are Euclidean similarities. A similastructure induces a holonomy representationHs of Γ = π1(S\p1, . . . , pn) into thesimilarity Sim(E2) C∗, where the tilde indicates the universal cover. We dethe dilatational holonomyas the induced representationHd :Γ → R, whereHd(γ ) =logdilatationHs(γ ) = log|Hs(γ )|. The rotational holonomycan almost be defined a


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Hr(γ ) = argHs(γ ), but for the slight complication that we need to take the argumeC∗, since we want to distinguish an angle of 4π from one of 2π. This notion of argument iswhat is used in the sequel. In particular, ifγ is a loop surrounding one of the distinguishpointspi , thenHr(γ ) is thecone angleatpi . In the case whereHd(Γ ) = 0, the similaritystructure is asingular Euclidean structure, with cone angles defined byHr as above. Inthe sequel, references to theholonomyof a similarity structure will actually mean thdilatational holonomyHd . A more concrete way to think of both similarity and Euclidestructure is by assembling our surface out of Euclidean triangles, in the pattern givsome complexT . In the case where the lengths of the edges of the triangles beingtogether agree, then we have a singular Euclidean structure. If not, then we have a simstructure. In either case, the vertices ofT are potentially cone points, with cone anggiven by the sums of the appropriate angles of the incident triangles.

Consider an oriented surfaceS, possibly with boundary, and assume thatS has aEuclidean metric (or, more generally, a similarity structure—with the exception oresults of Section 6, some of which depend on the metric Theorem 6.3, the mstructure or lack thereof plays no role inthe arguments). with cone singularities. Wwill be dealing withsemi-simplicialtriangulations ofS, that is, triangulations where tw(not necessarily distinct) closed cells might intersect in a collection of lower-dimenscells. All triangulations will be assumed semi-simplicial, unless specified otherwissubcomplexF of T will be calledclosedif whenever an open faceF is in F , so are all ofthe faces of∂F .

Assume now that the surfaceS is equipped with a finite geodesic semi-simplictriangulationT , such that the 0-skeleton ofT , which is denoted byV (t), contains allof the cone points ofS. Each face ofT is then a Euclidean triangle. There are two kindsedges ofT : the interior edges, incident to two faces ofT , and the boundary edges, incideto only one face ofT .

Definition 1.1. Let e be an edge ofT . First, suppose thate is a boundary edge, and lt = ABC be a face ofT incident toe, so thate = AB. Then thedihedral angleδ(e) at eis the angle oft at the vertexC. Now, assume thate is an internal edge ofT , so thate isincident tot1 = ABC andt2 = ABD, so thate = AB. Then the dihedral angle ate is thesum of the angle oft1 atC, and the angle oft2 atD. Theexteriordihedral angleδext(e) ate is defined to beδext(e) = π − δ(e).

Thecone angleat an interior vertexv of T is the sum of all the angles of the faces ofT

incident tov atv; theboundary angleat a boundary angle is defined in the same way.

The following is a slight extension of [20, Lemma 4.2].

Lemma 1.2. The cone angle at an interior vertexv ∈ V (T ) is equal to the sum of thexterior dihedral angles at the edges ofT incident tov. At a boundary vertex, the boundaangle is equal to the sum of the exterior dihedral angles as above, lessπ .

Proof. First, letv be an interior vertex. Suppose that there aren trianglest1, . . . , tn incidentto v. The sum of all of their angles isnπ. The cone angle atv is the sum of the angles of th


orem

geslarityn

tricar

trianglesti at v. On the other hand, the sum of the dihedral angles of the edgese1, . . . , en

incident tov is the sum of all of the angles ofti not incident tov. Thus,

nπ = Cone angle atv +n∑

i=1

δ(ei). (1)

The result follows by rearranging the terms.If v is a boundary vertex, and there aren faces incident tov, then there aren+ 1 edges,

and Eq. (1) becomes:

nπ = boundary angle atv +n+1∑i=1

δ(ei), (2)

and the result follows by rearranging terms, as above.Observation 1.3. The sum of all of the dihedral angles of all of the edges ofT is equal totheπ |V (T )|—in combination with the Lemma above this gives the Gauss–Bonnet thein this polyhedral context, since the curvature at an interior vertexv of T is defined to be

2π, cone angle atv,

while the curvature at a boundary vertex is defined to be

π, boundary angle atv.

Proof. Every angle of every face ofT is opposite to exactly one edge ofT . The next theorem is [20, Theorem 6.16]

Theorem 1.4. Let ∆ :E(T ) → (0,2π) be an assignment of dihedral angles to the edof T , and Hd a holonomy representation. There exists at most one singular simistructure onS with holonomyHd (and in particular, at most one singular Euclideastructure, up to scaling), such that so thatδ(e) = ∆(e), for every edgee ∈ E(T ).

Definition 1.5. A triangulationT with δ(e) π for every interiore ∈ E(T ) is called aDelaunay triangulation.

Let∆ be a map,∆ :E(T ) → (0,2π). When does there exist a singular Euclidean meon S with the dihedral angles prescribed by∆? It is clear that there are certain lineconstraints which must be satisfied—to wit, for every facet = ABC of T , we must be ableto find anglesα, β , andγ , such that:

Positivity. All angles are strictly positive.


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ls

ent

Euclidean faces.For every facet ,

αt + βt + γt = π.

Boundary edge dihedral angles.For every boundary edgee = AB, incident to the triangleABC,

γ = ∆(e).

Interior edge conditions.For every interior edgee = AB, incident to trianglesABC andABD,

α + δ = ∆(e).

The system of equations and inequalities above specify alinear programL. The feasibleregion (set of solutions) ofL must be non-empty in order for us to have any hope of haa singular Euclidean metric onS with prescribed dihedral angles. One of the princiresults (Theorem 6.1) of [20] is that if all of the dihedral angles are no greater thanπ (thatis, the triangulation is Delaunay), then Conditions 1–4 are also sufficient, thus:

Theorem 1.6. If the feasible region ofL is non-empty and every dihedral angle ismostπ , then there exists a similarity structure with any prescribed holonomyHd and,in particular, a singular Euclidean metric onS with the prescribed dihedral angles—thstructure is unique by Theorem1.4 (and the metric is unique up to scaling).

2. Necessary conditions on dihedral angles

In order for the linear programL to have any chance of having a solution, the diheangles∆ must satisfy some constraints. Indeed, supposeL has a solution.

Condition 2.1. All of the dihedral angles must be positive.

Furthermore, the sum of all of the dihedral angles of all of the edges ofT must be equato the sum of all of the angles of all of the faces ofT , or F(T )π . On the other hand, it ialmost equally obvious that ift1, . . . , tn ∈ F(T ) is some proper subset of the faces ofT ,then the sum of the dihedral angles at the edges incident to one of theti must be strictlygreater than thenπ . In other words:

Condition 2.2. Let F ⊆ F(T ), and letE(F) be the set of all edges incident to an elemof F . Then ∑

e∈E(F)

∆(e) π |F |, (3)

with equality if and only ifF = F(T ) or E(F) = ∅.


dge

f

he

the

Definition 2.3. For a subcomplexF of T , we define the excess ofF to be

excessF =∑

e∈E(F)

∆(e) − π |F |.

Definition 2.4. An edgee of a subcomplexF of T is a relative boundary edgeof F if itis incident to a top-dimensional face ofF on exactly one side, and is not a boundary eof T .

The following lemma will prove useful in the sequel.

Lemma 2.5. Let ∆ :E(T ) → (0,π] be an assignment of dihedral angles to edges oT

satisfying Condition2.2. Let t1, t2, . . . , tn be a collection of closed triangles ofT , and letF = t1 ∪ t2 ∪ · · · ∪ tn; assumeF = T . Let

∑∂ F be the sum of the dihedral angles of t

boundary edges ofF . Then

0 < excessF <∑∂

F . (4)

Proof. The first inequality of 4 is a restatement of Condition 2.2, applied tosubcomplexF . To show the second inequality, let

F =⋃

t∈F(T )\t1,...,tnt .

Evidently,F ∪F = T , while F ∩F = ∂F . Applying Conditions 2.2 toF , we see that

0 <∑

e∈E(F)

∆(e) − π∣∣F(F)

∣∣, (5)

while applying them toF , we see that

0 <∑

e∈E(F)

∆(e) − π∣∣F (

F)∣∣. (6)

Note now that ∣∣F(F)∣∣ + ∣∣F (

F)∣∣ = ∣∣F(T )

∣∣, (7)

while ∑e∈E(F)

∆(e) +∑

e∈E(F)

∆(e) −∑∂

F =∑

e∈E(T )

∆(e). (8)

Adding inequalities (5) and (6), and applying Eqs. (7) and (8), we obtain


linear

fore

d

l. Themumt thee

∑e∈E(F)

∆(e) − π∣∣F(F)

∣∣<

( ∑e∈E(F)

∆(e) − π∣∣F(F)

∣∣) +( ∑

e∈E(F)

∆(e) − π∣∣F (

F)∣∣)

=( ∑

e∈E(F)

∆(e) +∑

e∈E(F)

∆(e)

)− (

π∣∣F(F)

∣∣ + π∣∣F (

F)∣∣)

=∑

e∈E(T )

∆(e) +∑∂

F + π∣∣F(T )

∣∣ =∑∂

F ,

where the last equality is obtained by applying the equality case of Condition 2.2.

3. Sufficiency of Conditions 2.1 and 2.2

Somewhat surprisingly, the trivial Conditions 2.1 and 2.2 guarantee that theprogramL has a solution:

Theorem 3.1. Let T be a semi-simplicial triangulation of a surfaceS, possibly withboundary, let∆ :E(T ) → (0,π] be a function on the edges ofT , Hd a representationπ1(S\V (T )) → R. There exists a similarity structure with holonomyHd on S, with theDelaunay triangulation of(S,E) combinatorially equivalent toT , and with dihedralangles given by∆ if and only if Conditions2.1 and2.2 are satisfied.

We will use the Duality Theorem of linear programming to prove Theorem 3.1. Bestating the Duality Theorem we need to recall some notions: Alinear programL consists ofa collectionC(L) of constraints, which are linear equations or(non-strict) inequalities, antheobjective functionF(L), which is a linear function. The set of points inR

n satisfyingthe constraintsC(L) is called thefeasible regionof L, which is, by definition, a polyhedraregion, possibly unbounded, possibly not of full dimension, and possibly emptysolutionof the linear program is a point where the objective function attains an extre(which may be a maximum or a minimum). The value of the objective function asolution is theobjectiveof L. If the feasible region ofL is empty, the program is said to binfeasible. Now we can state the Duality Theorem:

Theorem 3.2 (Duality Theorem of Linear Programming).LetP be a linear program of theform:

Minimizec⊥x, subject to the constraintsAx = a, x 0.

Then the dual ofP is the programP ∗:

Maximizea⊥λ, subject to the constraintsλ⊥A c.


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The feasible region ofP is non-empty if and only if the objective ofP ∗ is finite.Conversely, the feasible region ofP ∗ is non-empty if and only if the objective ofP isfinite. If neither feasible region is empty, then the values of the objective functionsP

andP ∗ are equal.

Remark. While the primal programP in the statement of Theorem 3.2 appears toof a very special form, it is not hard to see that any linear program can be writtthis form. Indeed, if our linear program asked us to maximize the objective, we calways convert it to a minimization problem by multiplyingc by −1. If our programdid not require the variables to be non-negative, we can always replace a variablx byx+ − x−, wherex+ andx− are both required to be non-negative. If the program had sinequalities of the formai

⊥x a, or of the formai⊥x a we can first convert all the

inequalities of the first type to those of the second type by negation, and then cthem to equations by introducingslack variablesxi 0, and requiringai

⊥x + xi = a.

Similarly, any program can be made to look like the dual programP ∗ in the statement oTheorem 3.2.

ProgramL is almost in the primal form needed by the Duality Theorem, but fordifferences: there is no objective function, and we want the primal variables (the angthe triangles) to be strictly positive, rather than just non-negative.

For the moment, let us sweep these issues under the rug, by setting the objective fto be 0, and allowing the angles to vanish—it will be quickly apparent how to fix thinglater. Let the modified program beL1. The dualL∗

1 of L1 is the following:

The dual program. Maximize

F(u,v): π∑

t∈F(T )

ut +∑

e∈E(T )

∆(e)ve,

subject to the conditionsut + ve 0 whenevere is an edge oft .

Theorem 3.3. Assume that the conditions described in Conditions2.1 and2.2 are satisfied.Then the objective function ofL∗

1 is non-positive. It equals0 if and only if there is au, suchthatut = −ve = u, for all t ∈ F(T ), e ∈ E(T ).

Proof.

Observation 1. Note that if there is au as required in the statement of the Theorem,objective function is, indeed, equal to 0. This is nothing other than the equality caCondition 2.2.

Now, letu = min(u1, . . . , uF(T )). Letu(1)i = ui −u, and letv(1)

j = vj +u, for all valuesof the indices. The new variables are still feasible forL∗

1, and by Observation 1 above, th

transformation does not change the value of the objective. Furthermore, ifu(1)i = 0 for all i,

then the objective is non-positive, and is equal to zero only if all ofv(1) are equal to zero
j


ts

he

andg

r,last

as well—this is so, since all of thevj must be non-positive, and all of their coefficien

are positive, by Observation 2.1. Assume, then, thatu(1)t > 0 for t ∈ F (1); F (1) is aproper

subset ofF(T ) by construction.

Observation 2. Suppose now thatu(1)t = u, for all t ∈ F1. Then

F π∑

t∈F (1)

u −∑

e∈E(F (1))

∆(e)u < 0,

by Eq. (3).

Now, let u(1) = mint∈F (1)(u(1)t ), and let u2

t = u(1)t = 0 if t /∈ F (1), and otherwise

u(2)t = u

(1)t − u(1). Likewise,v(2)

e = v(1)e if e /∈ E(F (1), and otherwisev(2)

e = v(1)e + u(1).

This still leaves us in the feasible region ofL∗1 and strictly increases the value of t

objective (by Observation 2). The new nonzero setF (2) is a proper subset ofF (1), andwe can repeat this process. In the end, we will wind up with a feasible pointu(k),v(k),with u(k) = 0, where the value of the objective is non-positive (by Observation 1),strictly greater than the value of the objective atu,v (by Observation 2), thus completinthe proof.

The above theorem shows that the feasible region ofL1 is non-empty. In order to finda solution with strictly positive angles, we write our anglesαi asαi = βi + ε. We requireall of the βi to be non-negative, and our new objective is simply−ε. Call the resultingprogramL2. Its dualL∗

2 has the following form:

Second dual. Maximize

F(u,v): π∑

t∈F(T )

ut +∑

e∈E(T )

∆(e)ve,

subject to the conditionsut + ve 0 whenevere is an edge oft , and also

3∑

t∈F(t)

ut + 2∑

e∈E(T ), e/∈∂T

ve +∑

e∈E(T ), e∈∂T

ve −1.

Theorem 3.4. The optimal value of the objective ofL∗2 is strictly negative.

Proof. Suppose the contrary. Thenut = −ve = u, for someu, by Theorem 3.3. Howevein that case the last inequality ofL∗

2 is not satisfied, since the left-hand side of theconstraint ofL∗

2 vanishes. Indeed, it is equal to

u

(3

∑1− 2

∑1−

∑1

).

t∈F(t) e∈E(T ), e/∈∂T e∈E(T ), e∈∂T


2,

onsot4.3 ismain

ithout

dgee

gle (in

heges

l

Observe, however, that 3∑

t∈F(t) 1 counts each non-boundary edge with multiplicityand each boundary edgewith multiplicity 1.

4. Dihedral angles of Delaunay triangulations

In the preceding section it was shown that in order for the linear programL to havea solution, it is necessary and sufficient that the putative dihedral anglesδ(e) satisfy theinequalities 2.1 and 2.2. Below, we use these to derive another set of necessary conditi4.3, and show that under the additional assumption that the original dihedral angles do nexceedπ , these are equivalent to the inequalities 2.2. The virtue of the inequalitiesthat it is easier to interpret them geometrically. A special case of them is one of theresults of [21]—the connection will be explained in Section 6.

First, some definitions (these are the analogs of the definitions of Section 1 wreference to face angles)—as beforeT is a triangulation ofS, V (T ) is the set of verticesof T , E(T ) is the set of edges, andF(T ) is the set of faces. We assume that each ee ∈ E(T ) is given a weightδ(e) ∈ R. The weightδ(e) will be called the dihedral anglat e. The quantityδext(e) = π − δ(e) will be called theexterior dihedral angleat e.

If v is a vertex ofT , then thecone angleCv is defined to be

Cv =∑

e incident tov

δext(e),

while thecurvatureκv is defined to beκv = 2π − Cv.

Note. The cone angle of a boundary point is, thus, not the same as the boundary anthe language of Section 1), but smaller byπ .

Below, it will often be useful to talk of thePoincaré dualof T . Recall that this is thecomplexT ∗, such that the set of vertices ofT ∗ is in one-to-one correspondence with tset of faces ofT , the set of edges ofT ∗ are in one to one correspondence with the edof T —two vertices ofT ∗ are joined by an edges if and only if the corresponding faces ofT

share an edge, and finally the faces ofT ∗ correspond to the vertices ofT —the vertices ofa facev∗ of T ∗ correspond to the faces ofT incident to the corresponding vertexv.

Definition 4.1. A subcomplexF of T is closed, if whenever a cellt is inF , then so are alof the lower-dimensional cells incident tot .

Definition 4.2. Thetotal curvatureof a subcomplexF of T is defined as

K(F) =∑

v∈V (F)

κv.

Notation. Let F be a subcomplex ofT , and letE′(F) be the set of those edgese of T

which are not edges of faces ofT , but such that at least one endpoint ofe belongs toT .For each edgee of T , definenF (e) to be the number of endpoints ofe which belong toF .


t

at thenless

em

nlls us

For example, ife ∈ E(F), thennF (e) = 2; if e /∈ E(F) ∪ E′(F), thennF (e) = 0.

Theorem 4.3. For every non-empty subcomplexF of T the following are equivalent:

(a)∑

e∈E′(F) nF (e)δext(e) 2πχ(F) − K(F), with equality if and only ifF = T .

(b) Conditions2.1 and2.2 hold.

Proof. We will show that (b)⇒ (a); the converse is immediate.

K(F) =∑

v∈V (F)

(2π −

∑v∈e

δext(e)

)= 2π

∣∣V (F)∣∣ −

∑v∈V (F)

∑v∈e

δext(e). (9)

The last sum of Eq. (9) can be rewritten thus:

∑v∈V (F)

∑e incident tov

δext(e) = 2∑

e∈E(F)

δext(e) +∑

e∈E′(F)

nF (e)δext(e). (10)

Finally, using the definition ofδext(e), and combining Eqs. (9) and (10) it follows tha

K(F) = 2πχ(F) + 2

( ∑e∈E(F)

δ(e) − π∣∣F(F)

∣∣) −∑

e∈E′(F)

δext(e). (11)

Now, assume that the dihedral angles satisfy the inequalities 2.2. This means thmiddle term on the right-hand side of Eq. (11) is non-negative (and strictly positive uF = T ). 4.1. Some corollaries and refinements

Here are some easy consequences of Theorem 4.3:

Special case 1.F = T . Then, sinceE′(F) = ∅, we just get the Gauss–Bonnet theor(with curvature defined in terms of the dihedral angles).

Special case 2.The complement ofF is an annulus containing no vertices ofT —thiscorresponds to a simple cycle in the Poincaré dualT ∗. For every edge inE′(F),ne = 2. Theorem 4.3 just says that

∑e∈E′ δext(e) > 0.

Special case 3.For every edge inE′(F), ne = 1. This will hold in the case whereF is asubset ofT with a collar, and removing the edges inE′ separates the 1-skeletoof T into (at least) two connected components. In this case, Theorem 4.3 tethat

∑e∈E′(F)

δext(e) 2πχ(F) − K(F).


ectivenent can

.

d

e

k

a

fck

cted.

A drawback of both Conditions 2.2 and Theorem 4.3 is that they require their respinequalities to be checked foreverysubcomplex ofT in order to verify whether a giveassignment of dihedral angles is admissible. It is not hard to see that this requirembe weakened somewhat.

Let F be the subcomplex in question.

Observation 4.4. It can assumed that every edge inE(F) is an edge of at least one face

Observation 4.5. The 1-skeleton of the Poincaré dualF∗ of F can be assumeconnected—this is somewhat stronger than saying thatF is connected.

Observation 4.6. It can be assumed that the 1-skeleton ofF∗—the Poincaré dual of thcomplement ofF—has no isolated vertices.

It is straightforward to check all of the above observations.In the special situation whereδ(e) π for all e ∈ E(T )—that is,δ is Delaunay, one

can further assume that every facef of F is adjacent to at most one face ofF . Otherwise,adjoinf to F , to create a new complexF ′. This complex has one more face thanF , butits sum of dihedral angles is at mostπ greater than that ofF . Hence, it is enough to checthatF ′ satisfies the hypotheses of Theorem 4.3, or Conditions 2.2.

Definition 4.7. A simplesubcomplex ofT is a subcomplexF such that bothF andT \Fare connected.

Lemma 4.8. The complexC is simple.

Proof. By construction,C is connected. Also, every point ofT \C can be connected bypath to a point ofF . SinceF is assumed connected, the lemma follows.Theorem 4.9. To verify that Conditions2.1 and 2.2 (or equivalently, conditions oTheorem4.3(b)) hold for all subcomplexes ofT , it is necessary and sufficient to chethem for simple subcomplexes.

Proof. By Observation 4.5, it is enough to check the connected subcomplexes ofT . If sucha subcomplexF is simple, then we are done. Otherwise, its complement is not conneLet C be a connected component of the complement ofF .

ConsiderF ′ =F ∪ C. By assumption,F ′ = T . Now

( ∑e∈E(F)

∆(e) − π∣∣F(F)

∣∣) +( ∑

e∈E(C)

∆(e) − π∣∣F(C)

∣∣) −∑∂

C (12)

=( ∑

′∆(e) − π

∣∣F(F)′∣∣). (13)

e∈E(F )


itt on

n al

in thescutset

tely

sxistssll

By Lemma 2.5 (more precisely, a version for simple complexes),( ∑e∈E(C)

∆(e) − π∣∣F(C)

∣∣) −∑∂

C < 0, (14)

thus, ( ∑e∈E(F)

∆(e) − π∣∣F(F)

∣∣) >

( ∑e∈E(F ′)

∆(e) − π∣∣F(F)′

∣∣). (15)

Thus, in order to check that Condition 2.2 holds forF , it is enough to check thatholds forF ′. The proof of Theorem 4.9 is finished by the obvious induction argumenthe number of connected components of the complement ofF .

In the case whereT is a genuine simplicial complex (that is, two cells intersect ilower-dimensional cell), simple subcomplexes corresponds tonon-coterminous minimacutsetsof edges ofT :

Definition 4.10. A collection C of edges ofT is a cutset if removing the edges inCdisconnects the 1-skeleton ofT . A cutsetC is minimal if no proper subset ofC is a cutset.A cutsetC is coterminousif all of the edges inC are incident to the same vertex.

In other words, a minimal cutset corresponds to a separating simple curvePoincaré dualT ∗ of T (the curve need not be closed ifT has boundary). A coterminoucutset corresponds to a boundary of a face in the dual. The non-coterminous simplecorresponding to the subcomplexF is nothing other than the set of edgesE′(F) definejust before the statement of Theorem 4.3.

In the case where the surfaceS is a flat disk, Theorems 3.3, 1.6, 4.3, and 4.9 immediaimply:

Theorem 4.11. Let T be a triangulation of the disk, let∆ :E(T ) → (0,π] be anassignment of dihedral angles to the edges ofT , let V∂(T ) be the set of boundary verticeof T and letΛ :V∂(T ) → (0,π] be the assignment of boundary angles. Then, there ea collection of pointsp1, . . . , pV (T ) in the planeE2 whose Delaunay triangulation icombinatorially equivalent toT , has dihedral angles given by∆, and whose convex huis a polygon with angles given byΛ, if and only if:

• If v is an interior vertex ofT , then∑e∈E(T ) incident tov

(π − ∆(e)

) = 2π. (16)

• If v is a boundary vertex ofT , then∑e∈E(T ) incident tov

(π − ∆(e)

) + (π − Λ(v)

) = 2π. (17)


ve in

curve

tatedded

ition,onidean

n Cutldheorem.that thel sorts

edatewith

shead

of

r 7])

• If E is a non-coterminous minimal cutset corresponding to a simple closed curthe Poincaré dualT ∗ of T , then

∑e∈E

(π − ∆(e)

)> 2π. (18)

• If E is a non-coterminous minimal cutset not corresponding to a simple closedin T ∗, and E separates the boundary vertices ofT into two groupsv1, . . . , vk andvk+1, . . . , v|V∂ (T )|, then

∑e∈E

(π − ∆(e)

) +k∑

i=1

(Π − Λ(vi)

)> 2π. (19)

Remark. Theorem 4.11 is nothing but one of the main results (Theorem 0.1) of [21], sin a different language (withoutmentioning convex ideal polyhedra), so we have succeein deducing that theorem in a purely combinatorial way from the results of [20]. In addTheorem 4.9 is seen to be a direct generalization of [21, Theorem 0.1] to a characterizatiof dihedral angles of Delaunay triangulations of arbitrary, possibly singular, Euclsurfaces.

5. A network flow approach

There is an alternative way to prove Theorem 3.3 which uses the Max Flow–Mitheorem of network flow instead of the Duality Theorem of linear programming. It shoube noted that the difference between the twoarguments is largely superficial, since tproof of Theorem 3.3 can be seen to essentially prove the Max Flow–Min Cut theThere are two reasons to set up the question as a result on network flow. The first isproof (hopefully) becomes clearer and more intuitive. The second is that the speciaof linear programs that arise in the theory of network flow have been heavily analyzfrom the viewpoint of complexity, which will allow us to give a very satisfactory estim(Section 8) for the running time of an algorithm to determine whether a structureprescribed dihedral angles actually exists.

A networkis a directed (multi)graphN , with two distinguished vertices:s (the source)andS (the sink). Each edge ofN has a certaincapacity, which is a real number, which ian upper bound on the amount of the commodity which can flow from the tail to theof the edge. AcutsetC of N is a collection of edges, the removal of which leavess andS

in two different connected components ofN\C. Thecapacity of the cutsetC is simply thesum of the capacities of the edges comprisingC. The capacity of an empty collectionedges is, of course, 0.

The Max Flow–Min Cut theorem of network flow (see, for example, [27, Chaptesays the following:


owset

the

al sort,es:ing

three

1,n

ting

of the

Theorem 5.1 (Max Flow–Min Cut).The maximal amount of a commodity that can flfrom the sources to the sinkS of a networkN is equal to the capacity of the smallest cutin N .

This theorem can be proved in a number of ways. The interested reader can adaptproof of Theorem 3.3 to show Theorem 5.1.

To use Theorem 5.1 for our purposes, we need to set up a network of a specistarting with a triangulationT . The vertices of this network are divided into four classs, F(T ), E(T ), andS. The sources is connected to all of the vertices correspondto the faces ofT with edges of capacity 1. We call themedges of level1. Every vertexcorresponding to a facet is connected to the three vertices, corresponding to theedges oft by edges of capacity 1. These are edges of level 2. Finally, eache ∈ E(T ) isconnected to the sinkS by an edge of capacityδ(e). These are edges of level 3.

The following statement is self-evident.

Observation 5.2. There exists a solution of the linear programL1 if and only if themaximal flow through the above-constructed networkN1 is equal to|F(T )|.

Proof of Theorem 3.1. Consider a cutC of N1. This will have some edges at levelremoving which will disconnect a subsetF0 of the faces ofT from the source. It is thenot necessary to remove any edges of level 2 emanating fromF0. Let F(T )\F0 = F1. LetF2 ⊆ F1 be those facesf for which the cutset contains all three edges of level 2 emanafrom f . Finally, F3 = F1\F2. All of the edges of level 3 (indirectly) emanating fromF3must be in the cutsetC. These are precisely the edges corresponding to the edgessubcomplex ofT whose faces are inF3.

What is the capacity ofC? Evidently, it is equal to

F0 + 3F2 +∑

e∈E(F3)

δ(e).

If we want the flow throughN1 to beF , we must have

F0 + 3F2 +∑

e∈E(F3)

δ(e) F,

Or, noting thatF1 = F − F0,

3F2 +∑

e∈E(F3)

δ(e) F1.

In the special case whereF2 = 0, it follows that

∑δ(e) F. (20)

e∈E(F3)


ry

es

eights

nt

two

SinceF3 could have been any subset ofF , it follows that Condition 2.2 is necessa(this is, in any event, self evident).

On the other hand, if Condition 2.2 holds for any subcomplex ofT , substituting theinequality 2.2 into (5), we see that

c(C) F0 + 3F2 + F3 = F + 2F2 F. The above result is not quite what is required:the non-strict inequality (20) implies th

existence of a consistent assignment of angles to the faces ofT , but some of these anglemay be equal to 0. In order to have the angles strictly positive, we must modify the win the networkN1 as follows:

Modify the capacity of level 1 edges to be 1− 3ε; those of level 2 to be 1− ε, andthose of level 3 to be 1− 2ε. Call the resulting networkN2. The existence of a consisteassignment of angles to the faces ofT where all the angles are no smaller thanε isobviously equivalent to the maximal flow inN2 being equal toF(T )(1− 3ε).

Consider now a cutC in N2, with notation as before. The capacity ofC will be:

(1− 3ε)F0 + 3(1− ε)F2 +∑

e∈E(F3)

δ(e) − 2εE(F3). (21)

Assume thatF2 = 0. The Min Cut condition together with expression (21) gives:

(1− 3ε)F0 +∑

e∈E(F3)

δ(e) − 2εE(F3) (1− 3ε)F,

or ∑e∈E(F3)

δ(e) (1− 3ε)F3 + 2εE(F3).

If ∅ ⊂ F3 ⊂ F , thenE(F3) > 32F3, and so

∑e∈E(F3)

δ(e) > F3.

Suppose now thatδ(F ′) − F ′ ψ > 0, for all proper non-empty subsetsF ′ of F .Then expression (21) is no smaller than

(1− 3ε)F0 + 3(1− ε)F2 + F3 + ψ − 2εE(F3). (22)

The edges ofF3 can be divided into two classes: interior edges (those incident totriangles ofF3)—these numberEi(F3), and boundary edges ofF3—those incident to onlyone triangle. These numberE∂(F3). SinceF3 is a proper subset ofF , E∂(F3) > 0. Clearly,

2E(F3) = 3F3 + E∂(F3). (23)

The lower bound Eq. (22) can thus be rewritten as

c(C) (1− 3ε)F0 + 3(1− ε)F2 + (1− 3ε)F3 + ψ − εE∂(F3) (24)

= (1− 3ε)F + ψ + (1− 3ε)F2 − εE∂(F3). (25)


f

an be

l

ion ofextract

n

y

esiredbefore,

Theo-

Sinceε 13 (a triangle cannot have all angles greater thanπ/3), we get

c(C) − (1− 3ε)F ψ − εE∂(F3). (26)

In other words, if∑

e∈E(F ′) δ(e) − F ′ ψ for every∅ ⊂ F ′ ⊂ F , there is a solution othe linear programL1 with all face angles of all triangles no smaller thanψ/E(T ).

6. Delaunay triangulations of infinite sets of points

In this section we will show that Theorem 3.1 for singular Euclidean structures cextended without change to infinite locally finite complexes:

Theorem 6.1. Let T be an infinite but locally finite complex, and let∆ :E(T ) → (0,π].Then there exists a singular euclidean structure onT , with cone points at vertices ofT ,whose Delaunay triangulation is combinatorially equivalent toT , and whose dihedraangles are given by∆ if and only if each finite subcomplexF ⊂ T with E(F) = ∅ haspositive excess.

There are two ingredients in the argument. The first (Lemma 6.2) is an extensSection 3, the second (Theorem 6.3) is a geometric estimate which will enable us tothe necessary subsequences.

Lemma 6.2. LetT be a complex, and let∆ :E(T ) → (0,π]. Then there exists a Euclideastructure with cone angles at vertices ofT , and dihedral angles given by∆, except at theboundary edges ofT , where they are smaller than prescribed by∆ if the excess of ansubcomplexF of T , such thatE(F) = ∅ is positive.

Remark. Lemma 6.2 can be viewed as a relative version of Theorem 3.1.

Proof. The argument parallels very closely that of Section 3. The existence of the dstructure is, as before, equivalent to a negative objective of a linear program, and aswe set up a slightly simpler linear program first. To wit, the programL′ is:

Positivity. All angles are strictly positive.Euclidean faces.For every facet ,

αt + βt + γt = π.

Boundary edge dihedral angles.For every boundary edgee = AB, incident to the triangleABC,

γ + xe = ∆(e),

where the slack variables (see the comments following the statement ofrem 3.2)xe are also non-negative.


lese new

the.ve

at

lack

Interior edge conditions.For every interior edgee = AB, incident to trianglesABC andABD,

α + δ = ∆(e).

We relax the programL′ to a programL′1 by dropping the requirement that the ang

be strictly positive, make the objective 0, as before, and we see that the dual to thweakened linear programL′

1 is the following:

The dual program. Maximize

F(u,v): π∑

t∈F(T )

ut +∑

e∈E(T )

∆(e)ve,

subject to the conditions:

The inequalities ofL∗1. ut + ve 0 whenevere is an edge oft .

New inequalities.Whenevere is a boundary edge ofT , ve 0 .

Since the constraints of the above programL′∗1 are a superset of the constraints ofL∗

1,Theorem 3.3 still tells us that the objective function is maximized if there exists au, suchthatut = −ve = u, for all t ∈ F(T ), e ∈ E(T ). Now, this is not enough to guarantee thatobjective is zero, since the equality case of Condition 2.2 (whenF = T ) no longer existsSince the new inequalities requireu to be non-negative, it follows that for the objectifunction to equal 0,u must be 0.

Now, we follow Section 3 again, to define a programL′2 in the same way as before (th

is, since we want the angles to be strictly positive, we setαi = βi + ε, etc., and to alsodefine its dualL′∗

2 . By the same reasoning as before, it follows that the objective ofL′∗2 ,

and hence ofL′2, is negative. In fact, we can do more: we can also require all of the s

variablesxe to be strictly positive. The (yet another) new dual programL′∗3 will have the

form:

Third dual. Maximize

F(u,v): π∑

t∈F(T )

ut +∑

e∈E(T )

∆(e)ve,

subject to the conditions:

ut + ve 0 whenevere is an edge oft .

ve 0, whene is a boundary edge ofT .

3∑

t∈F(t)

ut + 2∑

e∈E(T ), e/∈∂T

ve + 2∑

e∈E(T ), e∈∂T

ve −1.


ry

rem of

ume,e

edt one

If we omit the last constraint, we remain with the dual programL′∗1 , and as the

discussion above showed, the objective of that can only be 0 ifut ≡ ve ≡ 0, which isat odds with the last constraint ofL′∗

3 . Theorem 6.3. Let p1, . . . , pn be a set of points in the plane, andD their Delaunaytriangulation. Assume that the shortest edge ofD has length1, and the excess of evenon-trivial subcomplex ofD is no smaller thand0. Then

diameterD (

4n

d0

)n

.

The proof of Theorem 6.3 will depend on a couple of easy auxiliary results:

Lemma 6.4. LetABC be a triangle witha/c, a/b < ε, 0 < ε < 1/10. Then,α < 2ε.

Proof. This is an immediate consequence of the Theorem of Cosines, or the TheoSines. Lemma 6.5. LetF be a subcomplex ofD. Then the excess ofF is∑

triangles ABC such that AB∈Fγ (ABC),

whereγ (ABC) is the angle atC.

Proof. This is immediate from the definition of excess.Proof of Theorem 6.3. Suppose that the conclusion of the theorem does not hold. Asswithout loss of generality, that the edge between the verticesp0 andp1 is the shortest on(and thus of length 1).

We construct a family of disksD1,D2, . . . ,Dn, all centered onp0, and such that thradius ofDi is equal to(4n/d0)

n. Let Ai = Di+1\Di , and letFi to be the maximal closesubcomplex ofD contained inDi . The hypothesis of the theorem ensures that at leasof the annuliAi contains no vertices ofD, let this annulus beAj . Then we claim that theexcess ofFj is smaller thand0. Indeed, consider a triangleABC of D adjacent toFj alongan edgeAB. The vertexC of ABC lies outsideDj+1, and thus the lengths ofAC andBC

are at least(4n/d0)j+1 − (4n/d0)

j , while the length ofAB is at most(4n/d0)j . Thus,

γ (ABC) <2

4n/d0 − 1<

d0

n,

by Lemma 6.4. Thus, by Lemma 6.5, it follows that

excessFj < d0,

contradicting the hypothesis of the theorem.


arethe 1-ch

n

icsuch

e

mplexe

es

’s

ous

t that

ombine

tn

Proof of Theorem 6.1. We only need to show that the positive excess conditionssufficient, since they are obviously necessary. In addition, we may assume thatskeleton of the Poincaré dualT ∗ is connected (if not, we prove the theorem for eaconnected component separately).

Now, we pick a pair of adjacent base verticesv1, v2 ∈ V (T ), and fixd(v1, v2) = 1. Now,for v,w ∈ V (T ) we definedc(v,w) to be equal to the combinatorial distance betweev

andw in the 1-skeleton ofT . Now, defineFi to be the span of all verticesu, such thatdc(v1, u) i. The complexFi is finite by local finiteness ofT . In addition,

⋃i Fi = T ,

so every finite subcomplex ofT belongs to someFi . For eachFi we consider a geometrrealizationS(Fi ), whose existence is guaranteed by Lemma 6.2 (there are manyrealizations, we pick any one of them).

Now, enumerate the faces ofT , in such a way thatt1 contains the edgev1v2, and, for anyj , the facestj andtj+1 are adjacent (that is, share an edge) inT . For each triangle, we havthe space of shapes (similarity classes), given (for example) by the complex parameterz,obtained by placing the first two vertices of the triangle at the points 0 and 1 in the coplane, and reading off the position of the third point (in theupper half-plane, assuming thtriangle is positively oriented). Theorem 6.3 tells us that for any facet of T , the set of shapparameters of realizations oft is contained in a compact setCt (since the ratio of lengthof any two sides is bounded by some constant, depending on the function∆). We can thinkof eachS(Fi ) as being an element ofC = ∏

j Ctj , which is a compact set by Tykhonovtheorem, and hence we can extract a convergent subsequence fromS(F1), . . . , S(Fk), . . . .Call the limit of that subsequenceS. Since the dihedral angles are obviously continufunctions of the triangle parameters, the dihedral angles ofS will be given by∆, and soSis the sought-after realization.Note. For the comfort of more analytically inclined readers, it should be pointed outhe last argument is a form Arzela–Ascoli.

6.1. Remarks and corollaries

Let us assume that the complexT has no boundary, and that the function∆ is suchthat all of the cone angles (computed using Lemma 1.2) are equal to 2π , so that anyrealization is Euclidean (and hence can be developed into the plane). Then, we can cTheorems 6.1 and 4.11 to get

Corollary 6.6. Let T be an infinite locally finite triangulation, and let∆ :E(T ) → (0,π]be an assignment of dihedral angles to the edges ofT , let then there exists a flasurfaceS, and a collection of pointsp1, . . . , pn, . . . in S whose Delaunay triangulatiois combinatorially equivalent toT and which has dihedral angles given by∆ if and onlyif:

• If v is an interior vertex ofT , then∑e∈E(T ) incident tov

(π − ∆(e)

) = 2π. (27)


e caseidealt

ingckingerolic,to be

edralo

stypeuthors,),

proofspect-

anner

s

nay

ection)

• If E is a non-coterminous minimal cutset, then

∑e∈E

(π − ∆(e)

)> 2π. (28)

The above corollary can be viewed as an extension of Theorem 0.1 of [21] to thof ideal polyhedra with infinitely vertices (and hence also of Andre’ev’s theorem forpolyhedra [2]), but not without certain caveats: even whenT is topologically a disk, it is noat all obvious whether the metric on the surfaceS (or, even,anysurfaceS) is geodesicallycomplete. If it is complete, it follows thatS is the Euclidean plane, and that the developmap is a global isometry, but otherwise the developing map is only an immersion. Cheif the given simply connected Euclidean surfaceS is the Euclidean plane is nothing oththan thetype problem—that is, we want to know whether a Riemann surface is parabhyperbolic, or elliptic. If we know the shapes of all the triangles, this can be shownequivalent to the recurrence of a random walk on the 1-skeleton of the complexT , wherean edgeAB, incident to trianglesABC andABD has weight 1/(cotC+cotD) [22]. Usingthis, it can be shown without too much difficulty that in the case where all of the dihangles are rational multiples ofπ , bounded away from 0 andπ , then this is equivalent tthe recurrence of the symmetric random walk on the 1-skeleton ofT .

In the special case of all dihedral angles being equal toπ or π/2, Corollary 6.6 reduceto an existence theorem for infinite locally finite circle packings. In this case, theproblem, and a number of others, has been studied at great length by a number of astarting with Koebe, but more recently by A. Marden (in the context of Schottky groupsW. Thurston, B. Rodin, and D. Sullivan, Z.-X. He, O. Schramm, and others.

It should be noted the Theorem 6.1 says nothing about uniqueness, and thecertainly does not show any form of uniqueness. In view of Theorem 1.4 one might suthat perhaps this could be shown with more work. In fact, it is quite clear from the abovementioned work on circle packing that uniqueness fails, though in a controlled mdescribed in the conjecture below:

Conjecture 6.7. Let T be a infinite locally finite complex,∆ a system of dihedral anglesatisfying the hypotheses of Theorem6.1. If there is a realization ofT supported on asimply connected domainΩ0 ⊂ C, then there is one supported oneverysimply connecteddomainΩ ⊂ C. Furthermore, such a realization is determined uniquely byΩ (up to thegroup of Möbius transformations fixingΩ).

7. Delaunay cells in moduli space

In Introduction we alluded to the cell decomposition of the moduli spaceM ofEuclidean structures, where the cells are given by Euclidean structures where the Delautriangulation has a fixed combinatorial typeT . Each cell is a convex polytopeP(T ), andthe results above can be used to describe the combinatorial and geometric structure ofP(T )

in some detail, although some interesting questions (discussed at the end of this sremain open.


etter

s

long

e

in theere

il inmore

lt ise, as

f

esal

ry, tocturenlideancan beration,ng

nd thenown

Consider a codimension 1 facef of P(T ). The facef may be either a boundary facof the moduli spaceM (viewed as a polyhedral complex) or an interior face. In the lacase,f corresponds to a change of combinatorial type of Delaunay triangulation fromT toT ′, and it is well understood that the primitive such change is given by adiagonal flip, sof corresponds to a cyclic quadrilateralABCD, where inT the Delaunay triangulation hatriangles (for example)ABC andCDA, while in T ′, the triangles areDAB andBCD.Onf , the quadrilateralABCD can be triangulated either way, but the dihedral angle a(either) diagonal is equal toπ .

Suppose now thatf is a boundary face ofM—in particular, this will mean that nonof the dihedral angles of the Euclidean structures in the interior off are equal toπ .By Theorems 4.9 and 4.3, it follows that there is a simple cocyclec∗ of T ∗ where theinequality 4.3(b) becomes an equality. This means, by the geometric estimatesbeginning of Section 6, that the collarC of c∗ is becoming long and thin (that is, thconformal modulus ofC diverges to∞), and sof corresponds to the Euclidean structupinching off along the curvec∗. The following construction (discussed in greater detaan upcoming paper of the author) helps visualize this pinching off in terms of theusual degeneration of hyperbolic surfaces:

Consider a triangulated singular Euclidean surface(S,E,P ). The data given by(S,E,P ) (initially using a triangulation, though it is easy to show that the resuindependent of triangulation) can be used to construct a cusped hyperbolic surfacfollows:

(1) To each trianglet of the triangulationT of (S,E,P ), associate an ideal triangleh(t).(2) For each pair of adjacent trianglest1 = ABC andt2 = ABD of T , we have the log o

the modulus of the cross-ratio of the four corresponding points:

r(t1, t2) = log|AC||BD||BC||AD| .

(3) For each pair of adjacent trianglest1 and t2 as above, glue the hyperbolic trianglh(t1) andh(t2) along the edge corresponding toAB with shear (see, e.g., [19]) equto r(t1, t2).

It is not hard to see that if we start with a Euclidean structure(S,E,P ), we will wind upwith a completecusped hyperbolic structure (actually, it is sufficient, but not necessastart with a Euclidean structure—some similarity structures will give a complete strualso). The construction thus defines a map (certainly not injective, but which can be showto be surjective using the construction of [8,14,16]) between the moduli space of Eucstructures with cone points and that of complete finite-area hyperbolic structures. Itshown (this was an important part of [17,21] for the case of genus 0) that the degeneas above, of the Euclidean structure on(S,E,P ), corresponds precisely to the pinchioff along a simple closed curve of the hyperbolic structure on(S,H(E),P ).

Remark. Another surjection between the space of singular Euclidean structures aspace of cusped hyperbolic structures is well-known, and is discussed in the well-k


by

fhat

gesyre

uce toglesoth

near

ual toxity

entoations

edly

al

, thehees

the

paper of Troyanov [25]. The two surjections arenot the same—this was remarkedC.T. McMullen.

8. Computational complexity

Consider the following two decision problems:

Problem 1. Let T be a simplicial complex, homeomorphic to a surfaceS (possibly withboundary), and letα :V (T ) → R

+ be an assignment of cone angles to the vertices oT .Does there exist a Euclidean structureE on S with the prescribed cone angles, such tthe Delaunay triangulation of(S,E) is combinatorially equivalent toT ?

Problem 2. Let T be a simplicial complex, homeomorphic to a surfaceS (possibly withboundary), and let∆ :E(T ) → (0,π] be an assignment of dihedral angles to the edof T . Does there exist a singular Euclidean structureE on S, such that the Delaunatriangulation of(S,E) is combinatorially equivalent toT , and whose dihedral angles agiven by∆.

By Theorem 1.6, there are efficient algorithms for both problems, since they redthe linear programL of Section 1. If the angles (cone angles in Problem 1, dihedral anin Problem 2) are rational multiples ofπ , such that the numerator and denominator are bbounded byC, then (by now) standard interior point methods allow us to solve the liprogramL usingO(n4(1 + logC)) arithmetic operation, each involving arithmetic usingprecisionO(n(1+ logC)). In the case where all of the prescribed cone angles are eq2π , the logC can be disposed with, and we wind up with an algorithm of bit-compleO(n5 log2 n).

Remark. In practice, the simplex algorithm appears much more efficient, and this has beused by M.B. Dillencourt to analyze all planar triangulations of up to 14 vertices, anddetermine which of them are combinatorially equivalent to planar Delaunay tessell[4,5].

For Problem 2, the network flow formulation of Section 5 turns out to give a marksuperior complexity. Indeed, it has been shown in [1] that for a network withn nodes,marcs, and the (integer) capacity of each arc bounded byU , we can determine the maximflow in time bounded byO(nm log((n/m)(logU)1/2 + 2)).

For the networkN1 of Section 5, assuming the genus of the surface is fixednumber of arcs and nodes in the network are both bounded by constant multiples of tnumberF of faces in the complexT . If all of the dihedral angles are rational multiplof π , with numerators and denominators all bounded in absolute value byC, the quantityU is bounded byCO(F) (since we need to compute the least common multiple ofdenominators), giving a running time bound ofO(F 5/2). For the programN2, the boundis the same, since the only difficulty consists of picking the right value ofε, and this canbe made to be 1/least common denominator of the dihedral angles.


canfs

:

sn

d

oneand to

tex.

9. Some applications to combinatorial geometry

The well-known theorem of Steinitz says that every three-connected planar graphbe realized as the 1-skeleton of a convex polyhedron inR

3, while a famous theorem oAleksandrov states that every Euclidean metric onS2 with positively curved cone-pointcan be realized as the induced metric on the surface of a convex polyhedron inR

3. Below,we show a negative result, which should be compared with the final example of [18]

Theorem 9.1. There exist infinitely many triangulations ofS2 which cannot be realized aa Delaunay triangulation with respect to the cone-points of any Euclidean metric oS2

with positively curved cone-points.

First a definition:

Definition 9.2. Let T be a triangulation. Thestellations(T ) of T is the complex obtaineby replacing each faceABC of T by three facesAOB, AOC, andBOC.

Theorem 9.1 follows immediately from the following claim.

Claim 9.3. Let T be any triangulation ofS2 with at least eight faces. Then the stellatis(T ) ofT is not combinatorially equivalent to the Delaunay triangulation of any Euclidmetric onS2 with positively curved cone-points, where the cone-points corresponvertices ofs(T ).

Proof. Let anold vertex ofs(T ) be one that was already a vertex ofT , while anewvertexbe one that was added at stellation. The setN of new vertices ofs(T ) corresponds tothe set of faces ofT . For any vertexv, recall thatC(v) denotes the cone angle atv. TheGauss–Bonnet theorem tells us that

∑v∈V (s(T ))

(2π − C(v)

) = 4π. (29)

Or, recombining the terms:

∑v∈V (s(T ))

C(v) = 2π(∣∣V (

s(T ))∣∣ − 2

). (30)

Note that every edge ofs(T ) is incident to an old vertex, and to at most one new verCombining this observation with Lemma 1.2, we see that for anyDelaunaytriangulationcombinatorially equivalent tos(T ), it must be true that

∑C(v)

∑C(v). (31)

v∈old vertices ofs(T ) v∈new vertices ofs(T )


r’s

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n

llybina-arest the

callic

eryven if

lationof

s).

rtices

m of

:

Combining Eq. (31) with (30), it follows that

∑v∈old vertices ofs(T )

C(v) π(∣∣V (

s(T ))∣∣ − 2

). (32)

Note now that|V (s(T ))| = |V (T )| + |F(T )|. By a standard computation using Euleformula for triangulations of the sphere, we know that|V (T )| = 1

2|F(T )| + 2, thus|F(T )| = 2|V (T )| − 4. Thus,∣∣V (

s(T )) − 2

∣∣ = 3∣∣V (T )

∣∣ − 6. (33)

Equations (33) and (32) together imply that theaveragecone angle at an old vertex os(T ) is at leastπ(3|V (T )| − 6)/|V (T )| = 3π − 6/|V (T )|. If |V (T )| > 6 it follows thatthe average cone angle at an old vertex is greater than 2π , which contradicts the assumptiothat the cone angles were positively curved.

10. Linear hyperbolic structures on 3-manifolds

As explained in [20], the study of Euclideantriangulations on surfaces is essentiaequivalent to the study of hyperbolic ideally triangulated complexes, which are comtorially just cones over the triangulated surface. Hence, the contents of this sectionclosely related to the subject-matter of much of the rest of the paper, in more than julinear programming approach.

Consider a 3-manifoldM3 with boundary a collection of tori, and consider a topologiideal triangulationT of M3. We would like to know when there is a complete hyperbostructure onM3, such thatT is a geometric ideal triangulation. In general, this is a vdifficult question, at least as hard as Thurston’s hyperbolization conjecture (since eM3 admits a hyperbolic structure of finite volume, there might not be an ideal triangucombinatorially equivalent toT ). However, below we will consider a “linear” versionthe question above.

Recall that an ideal simplexS in H3 has the properties that

Euclidean links.The link of each vertex is a Euclidean triangle.Equal opposite dihedral angle.If S = ABCD, then the dihedral angles at the edgesAB

andCD are equal (this is actually a consequence of the condition on the link

An ideal simplex is thus determined by the angles of the link of any one of its ve(all links are easily seen to be the same).

Now, if T comes from a genuine hyperbolic structure, it must be true that the suthe dihedral angles incident to the edges ofT must equal 2π , and so forT to correspondto such a structure, the following linear program must have a strictly positive solution

The variables.These are the dihedral angles ofthe simplices. For each simplexS we usethree anglesα,β, γ corresponding to the angles of the link of one vertices ofS.


es

, thena

l, noionsgtence

elyybeing

In-ic.

f theles

ive of

ceer

rr

Simplex conditions.For each simplexS, the sum of the dihedral angles is equal toπ :

fS : α + β + γ = π. (34)

Edge conditions.For each edgee of T the sum of the dihedral angles of all simplicincident toe equals 2π :

fe:∑

λi = 2π. (35)

Definition 10.1. We say that if the above linear program has a non-negative solutionthe pair(M3, T ) admits aweak linear hyperbolic structure. If the linear program hasstrictly positive solution, then we say that the pair(M3, T ) admits alinear hyperbolicstructure.

It is clear that the existence of a linear hyperbolic structure is, in generaguarantee of the existence ofa genuine hyperbolic structure, since the linear conditdo not preclude “Dehn surgery” singularities, aswell as translational singularities alonedges ofT (see [15,26] for discussion). Conversely, as remarked above, the exisof a complete hyperbolic structure onM3 is no guarantee that there is a positivoriented ideal triangulation combinatorially equivalent toT (or, indeed, any positiveloriented triangulation). However, linear hyperbolic structures have the advantage ofconsiderably more tractable.

Theorem 10.2. In order for there to exist a weak linear hyperbolic structure for(M3, T ),every normal surface with respect toT must have non-negative Euler characteristic.order for there to exist a linear hyperbolic structure for(M3, T ), every non-boundaryparallel normal surface with respect toT must have strictly negative Euler characterist

Proof. We will use the method of Section 3. First, let us write the dual program olinear program (referred to asLh in the sequel) for weak hyperbolic structure: Our variabarevS, whereS ranges over all the simplices ofT , andve whereS ranges over all theedges ofT .

The dual programL∗h is:

• Maximize∑

S∈S(T ) vS + 2∑

e∈E(T ) ve .• Subject tovS + ve1 + ve2 0 for all facesS and pairs of opposite edgese1, e2.

In order for the primal program to have a non-empty feasible region, the objectthe dual must be non-positive.

Consider now a normal surfaceS (see, e.g., [9,10,13] for rudiments of normal surfatheory). The surfaceS intersects each simplexS in a collection of disks, which arcombinatorially either triangles (cutting off one vertex ofS from the other three), oquadrilaterals (separating one pair of vertices from another). For each simplexS, definetS(S) to be the number of triangular components ofS ∩ S, andqS(S) to be the numbeof quadrilateral components. For each edgee of T , define ie(S) to be the numbe


h

t

traints

d

, and

les

nyus

new

of intersections ofS with e. The intersections ofS with the simplices ofT inducesa triangulationT of S, where each triangular disk contributes one triangle, and eacquadrangle contributes two. DefineuS(S) to be the number of triangles ofT sitting insidea simplexS. Evidently,uS = tS + 2qS .

Note that, by Euler’s formula,

χ(S) =∑

e∈E(T )

ie − 1

2

∑S∈S(T )

uS.

This is seen to be very similar in form to the objective function ofL∗h, so let us se

vS = −uS , andve = ie.

Lemma 10.3. The assignment of the variables as above satisfies the inequality consof L∗

h.

Proof of Lemma 10.3. We need to check that forS a simplex ande1, e2 a pair of disjointedges ofS. We need to check that

ie1 + ie2 uS. (36)

By linearity, we need just check the inequality (36) for connected components ofS ∩ S. Ifthat component is a trianglet , thent contributes 1 toie1 + ie2 (since a “normal triangle”intersects exactly one of each pair of opposite edge). Also,t contributes 1 touS , so for atriangular face, the right- and left-hand sides of (36) are equal.

Suppose now we have a quadrilateral componentq . It contributes 2 to the right-hanside of (36). As for the left-hand sides,q hits two pairs of opposite sides ofS, so if e1ande2 is one of those pairs, then we have a contribution of 2 to the left-hand sideotherwise we have a contribution of 0.Remark 10.4. Notice that ifS is such that all of the components ofS ∩S, for all S ∈ S(T )

are triangles, then all of the constraints ofL∗h are equalities with the assignment of variab

as above. Any suchS is easily seen to be a union of boundary tori.

Lemma 10.3 concludes the proof of the “weak” part of Theorem 10.2, since if aShad positive Euler characteristic, the programL∗

h would have a positive objective, and ththe programLh would have no solution.

For the proof of the “strong part” we use the same trick as in Section 3. Definevariablesα′ = α + ε, etc. Our primal linear hyperbolicity programLs is now:

• Minimize −ε.• Subject to face constraintsα′ + β ′ + γ ′ + 3ε = π.

• And to edge constraints∑

α′ +v(e)ε = 2π , wherev(e) is the valence ofe.

The dual programL∗s is then:


e.

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incaréat is,

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J.

1970)

(1)

91–

565.n-

York,

• Maximize∑

S∈S(T ) vS + 2∑

e∈E(T ) ve .• Subject to the old constraintsvS + ve1 + ve2 0 for all facesS and pairs of opposite

edgese1, e2.• And the new constraint 3

∑S∈S(T ) vS + ∑

e∈E(T ) v(e)vE −1.

In order for(M3, T ) to be linearly hyperbolic, the objective must be strictly negativObserve that the sum of the left-hand sides of the old constraints isequalto the left-

hand side of the new constraint. Indeed, eachvS occurs three times (once for each pairopposite edges), and eachve occurs the number of times equal to the valence ofe. Hence,the new constraints simply says that in at least one of the old constraints the ineqmust bestrict. Keeping in mind Remark 10.4, this implies that every non-boundary-panormal surface must have strictly negative Euler characteristic, thus proving the secoof Theorem 10.2.

Some remarks may be in order: It is easy to see (and not surprising) that an idtheorem can be proved if the cone angles around the edges ofT are required to not bsmaller than 2π , while if the angles are smaller than 2π , one can show an analogo“orbifold” version of the theorem. A more interesting question is whether the converTheorem 10.2 holds. This is equivalent to asking whether every assignment of vasatisfying the constraints of programsL∗

h andL∗s comes from a normal surface.

Acknowledgments

The author thanks the École Normale Superieure de Lyon and the Institut Henri Pofor their hospitality. The special case of Theorem 3.1 for Euclidean structures (thtrivial holonomy) has been independently shown for closed surfaces by Veech [28] aBowditch [7] using completely different methods. Theorem 10.2 is due to A. Casson

References

[1] K. Ahuja, J.B. Orlin, R.E. Tarjan, Improved time bounds for the maximum flow problem, SIAMComput. 18 (1989) 939–954.

[2] E.M. Andreev, On convex polyhedra of finite volume in Lobachevskii space, Math. USSR-Sb. 12 (255–259.

[3] A. Casson, private communication.[4] M.B. Dillencourt, private communication.[5] M. Dillencourt, Polyhedra of small order and their Hamiltonian properties, J. Combin. Theory Ser. B 66

(1996) 87–122.[6] B.H. Bowditch, D.B.A. Epstein, Natural triangulations associated to a surface, Topology 27 (1) (1988)

117.[7] B.H. Bowditch, Singular Euclidean structures on surfaces, J. London Math. Soc. 44 (3) (1991) 553–[8] D.B.A. Epstein, R.C. Penner, Euclidean decompositions of non-compact hyperbolic manifolds, J. Differe

tial Geom. 27 (1988) 67–80.[9] W. Haken, Theorie der Normalflächen, (German), Acta Math. 105 (1961) 245–375.

[10] G. Hemion, The Classification of Knots and 3-Dimensional Spaces, Oxford University Press, New1992.


ent.

ex

n

n,

87)

nt.

andaa,

).

r,

28)

.78,

[11] J. Harer, The virtual cohomological dimension of themapping class group of an orientable surface, InvMath. 84 (1) (1986) 157–176.

[12] C.D. Hodgson, I. Rivin, W.D. Smith, A characterization of convex hyperbolic polyhedra and of convpolyhedra inscribed in the sphere, Bull. Amer. Math. Soc. 27 (3) (1992).

[13] W. Jaco, Lectures on Three-Manifold Topology, in:CBMS Regional Conf. Ser. in Math., Vol. 43, AmericaMathematical Society, Providence, RI, 1980.

[14] M. Näätänen, R.C. Penner, The convex hull construction for compact surfaces and the Dirichlet polygoBull. London Math. Soc. 6 (1991) 568–574.

[15] W.D. Neumann, D. Zagier, Volumes of hyperbolic 3-manifolds, Topology 24 (3) (1985) 307–332.[16] R.C. Penner, The decorated Teichmüller spaceof punctured surfaces, Comm. Math. Phys. 113 (2) (19

299–339.[17] I. Rivin, C.D. Hodgson, A characterization of compact convex polyhedra in hyperbolic 3-space, Inve

Math. 111 (1) (1993).[18] I. Rivin, On geometry of convex ideal polyhedra in hyperbolic 3-space, Topology 32 (1) (1993).[19] I. Rivin, Intrinsic geometry of convex ideal polyhedra in hyperbolic 3-space, analysis, algebra,

computers in mathematical research, in: Proceedings of the Nordic Congress of Mathematicians, Lule1992, in: Lecture Notes in Pure and Appl. Math., Vol. 156, Dekker, 1994.

[20] I. Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (3) (1994[21] I. Rivin, A characterization of ideal polyhedra in hyperbolic 3-space, Ann. of Math. 143 (1) (1996).[22] I. Rivin, Personal communication.[23] J. Steiner, Systematische Entwicklung der Abhängigkeit Geometrischer Gestalten von Einander, Reime

Berlin, 1832, appeared in J. Steiner’s Collected Works, 1881.[24] E. Steinitz, Über isoperimetrische Probleme bei konvexen Polyedern, J. Reine Angew. Math. 159 (19

133–143.[25] M. Troyanov, Les surfaces euclidiennes à singularites coniques, Enseign. Math. 32 (1–2) (1986) 79–94[26] W.P. Thurston, Geometry and topology of 3-manifolds, Unpublished Princeton Lecture Notes for 19

distributed by MSRI, Berkeley.[27] J. van Lint, R.M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 1992.[28] W. Veech, Delaunay partitions, Topology 36 (1) (1997) 1–28.

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