Pacific Journal ofMathematics

ON l-SIMPLICIAL CONVEXITY IN VECTOR SPACES

TUDOR ZAMFIRESCU

Vol. 22, No. 3 March 1967

PACIFIC JOURNAL OF MATHEMATICS

Vol. 22, No. 3, 1967

ON /-SIMPLICIAL CONVEXITY IN VECTOR SPACES

TUDOR ZAMFIRESCU

The paper is concerned with a generalized type of con-vexity, which is called /-simplicial convexity. The name isderived from the simplex with I vertices, an Z-simplicialconvex set being the union of all (ΐ — l)-simplexes withvertices in another set, i varying between 1 and I. The basicspace is a linear one.

For convex sets the Z-order (which is a natural numberassociated to an ί-simplicial convex set) is a decreasing func-tion of I. Several inequalities between I- and ά-orders areestablished. In doing this the case of a convex set and thatof a non convex set are distinguished.

Besides these inequalities, the main result of the paper isthe proof of non monotonicity of the border, given by anexample in a 34-dimensional linear space.

Let us consider a real vector space and recall some notations anddefinitions, given in [2].

The convex cover (hull) of a set M is gf(M). We denote&({xu , xs}) by &(xlf , x8), xlf , xs being not necessarily distinctpoints in the space.

The operation £^, called Z-simplicial convex cover, and defined by

Sφί) = {U if (^, . . . , ^ ) : f t G J ί , l ^ i ^ l }

(for arbitrary set M and natural number I ^ 2) will play a significantrole throughout the paper. The operation S^ is studied in [1], whereis denoted by conz. It is easy to verify the following elementaryproperties of this operation:

(l) ^ ς Λ = ^ ς o ^ ; ,

(2) £smP = ^ 1 , and

(3) &>m(M)a&(M),for arbitrary m, n, p ^ 2 and M. S^fM) is an increasing1 functionof I. It is also an increasing function of M with respect to the in-clusion ordering. Let us denote S^m({xu , xs}) by S^m{xu , xs).

A set K is said to be Isίmplicίal convex if there exists a subsetM such that K = SφΛ).

The l-order (l-simplίcial convexity order) of an i-simplicial con-vex set K is

ωt(K) = supmin {k: Sf\(M) = K} .M

1 By increasing and decreasing functions we mean here not necessarily strictlyincreasing and strictly decreasing ones.

565

566 TUDOR ZAMFIRESCU

A set K is said to be simplicial convex if there exists a numberI such that K is ϊ-simplicial convex.

The degree of a simplicial convex set K is

δ(K) = min {I: K is i-simplicial convex} .

The order {simplicial convexity order) of a simplicial convex setK is

= sup ωι(K) .I

The power of a simplicial convex set K of finite order is

Δ(K) = min {I: O(K) = ω,(iί)} .

It is proved in § 1 of both [1] and [2] that

gf (M) = <9ϊίl08ιnl+1(M)

in an π-dimensional vector space.

2* Relations between Z-simplicial and Zosimplicial convexityorders* In this section some inequalities concerning ί-order and k-order of a same set will be established.

THEOREM 1. The l-order and k-order of a convex set K satisfythe inequalities'.

[log, (ZV- 1 + 1)] ωk(K) [log, (W*> - 1)1 + 1 .

Proof. Consider a set M such that

Sφί) - K .

If k< l\

K - SftM) c ^

S i n c e i f i s c o n v e x t h e i n v e r s e i n c l u s i o n h o l d s t o o a n d K = S ^ ( )L e t x e K . S i n c e

^ωι{K)(M) = K ,

x belongs to a simplex with vertices xl9 , xs e M(s ^ W^)- Thereexists a linear manifold of dimension lωι{K) — 1 containing g*(ίc,, , a?β).Following the last remark of §1,

%>(xl9 . . . , » . ) = ^ ^ * ( I d > i ( Λ : > - 1 ) ^ 1 ( a ? 1 > , a?.) c

h e n c e

ON Z-SIMPLICIAL CONVEXITY IN VECTOR SPACES 567

The inverse inclusion holds owing to the convexity of K, whence

ω{K) = K .

Thus

ωk(K) £ [log, (ί«ι<*> - 1)] + 1 .

The symmetry in I and k gives

ω^K) ^ [log, (£«*<*> - 1)] + 1 ^ log, (£«*<*> - 1) + 1 ,

i.e.

It follows that

[log, (W^- 1 + 1)] ^ log, (IV*)-1 + 1) ^ ωΛ(X)

and both inequalities are obtained.

In [2], we have established that in general, λ -simplicial convexitydoes not imply ϊ-simplicial convexity for I < k. However, if k is apower of I this implication holds.

THEOREM 2. For all natural numbers k, I, q Ξ> 2 satisfyingk = lq, non convexity and k-simplicial convexity imply l-simplicialconvexity; also the k-simplicial and l-simplicial convexity orders verifythe inequalities

qωk ^ ωι ίg qωk + q — 1 .

Proof. Let M be such that

= C ,

where C is a non convex, fe-simplicial convex set. Then

whence C is Z-simplicial convex and

To prove the other inequality, suppose that there is a set Mr

such that

^ m ( M ' ) = C

with

568 TUDOR ZAMFIRESCU

w& ^ Qθ)k(C) + q .

Then, either m — qo)k(C) + g and

or m > qωk(C) + g and

both impossible. Following Theorem 5 of [2],

ω,(C) = sup {m: ^m(M) = C} .M

Hence

ω,(C) ^ gωΛ(C) + q - 1 ,

which concludes the theorem.A different inequality is obtained, if k = i9, for a convex set.

We have, indeed, by Theorem 1, ωk ^ log* (iωί — 1) + 1, i.e.

and ωk ^ logfc (ίωr 2 + 1), i.e.

k^ ^ ί ' r 1 + 1 .

Hence

whence

q(ωk - 1) < ωι

and

iffω* ^ lωrι + 1 ,

whence

qωk > ωι - 1 .

Therefore

These inequalities and those of Theorem 2 show that, for arbi-trary sets, A -simplicial convexity implies Z-simplicial convexity (fork = lq) and

qωk — q + 1 <; ωt ^ qωk + q - 1 .

ON Z-SIMPLICIAL CONVEXITY IN VECTOR SPACES 569

3* M o n o t o n i c i t y of o)t(K) for c o n v e x K. I t is proved in § 4

of [2] t h a t ωk ^ ωι if k is a multiple of I. Moreover, we shall prove

t h a t for convex K, ωk(K) ^ ωx(K) if k ^ I.

THEOREM 3. For a convex set K, the l-simplicίal convexity orderis decreasing on I and Ω(K) = ω2(K).

Proof. Prove that ω^K) is a decreasing function of I. Suppose,on the contrary, that i > j and

ωά{K) .

It follows that

ωt(K) - 1 ^ ωά{K)

and

But, from Theorem 1,

ωάK) ^ log, (j"J^ - 1) + 1 ,

whence

/j/ωi(K)-ί < jo)j(K) _ ]_ ^ jo)j(K)

The contradiction shows that ωz(if) is decreasing of I, for convexK. Since iΓ is 2-simplicial convex,

Ω(K) - ωδ(iC)(iί) = ω2(K) .

4* Non monotonicity of o)ι% It may be conjectured that ωt isin general a decreasing function of i, i.e. ωz(C) is also decreasingfor non convex C. Then the inequality of Theorem 11 would betrivially implied by Theorem 6, both of [2], and Ω and δ would equalrespectively ωs and A.

On the other hand one can believe that Theorem 2 can be obtainedfrom two more general inequalities, for non convex sets, like

(*) [log, (lω* + 1 ) 1 - 1 ^ 0 ) * ^ [log, (1-1+1 - 1)]

in the same way as, for convex sets,

qωk — q + 1 ^ ωι <; qωk (k = lq)

is implied by Theorem 1.It should be noted that each inequality of (*) would imply that

ωι is decreasing. If, on the contrary i < j and ω{ < ω3Ί then

570 TUDOR ZAMFIRESCU

which contradicts the two inequalities

and

ωό < [log, (i^1 - 1)]

Also,

iωi+1 < jωj

contradicts the two inequalities

jωj + l <; ίωi+1

and

The conjecture that o)ι is, in general, a decreasing function ofI (and with it also relation (*)) is disproved by the following counter-example.

PROPOSITION. Let 7 be a 34-dimensional real vector space andC be the union of the subsimplexes with 18 vertices of a simplex with35 vertices in 5Γ Then C is both 2-simplicial convex and 3-simplicialconvex, ω2(C) = 1 and ω3(C) = 2. Also Ω(C) = 2 and Δ{C) = 3.

Note. This proposition and its proof, are the simplest that wecould find to provide our counter-example. We ask for simpler ones.In fact, we have found such a simpler example in a vector space ofsmaller dimension, but the proof was much more complicated. How-ever it should be interesting to find the smallest dimension of thespace in which such an example can be found, even if its proof isdifficult.

Proof. Let S)(i = 1, , (f)) be the subsimplexes with j vertices(j = 2, , 34) of the given simplex S. Thus

c = u sι8.i

(1) The 2-simplicial and 3-simplicial convexity of C follow from

c =

(2) Prove that ω2(C) = 1. Suppose that 6^\M) = C.(a) First, we shall prove that

ON Z-SIMPLICIAL CONVEXITY IN VECTOR SPACES 571

Suppose there exists

then y belongs to the interior of a simplex S)a with 10 ^ j ^ 18.(a) If 10 ^ i ^ 17, let Sjj-i be the simplex with 35 - j vertices

disjoint from S)a. If <Pζ(M) contains also a point z in the interiorof a subsimplex S]-j of S^-y, with ί ^ 19, then &(y> z) intersectsthe interior of gf (S*α U S\ί.ά), which is impossible; hence

ΓΊ Silj c {U SlLj: S ^ - c Sjt,} c { u S j : S 5 c Sβ_,.}

for 10 ^ j ^ 16 or

^t(M) n sii c {a?lf..., xw] n s;8

b c {u si: s^ c si8

δ}

and if

o 1 8 L_ o35_j ,

then

Sζ(M) ΓΊ Sί8

rf c {U Si: S i c Stlj} Π Sί ί - { U S j S j c Sit} .

It follows t h a t

^2\M)n s; 8

d c{us; 6 :s j 6 cs; 8

d } ^ s;8

d,

whence

sit,

absurd.(b) If j — 18, then the segment joining y with a vertex &% of S

that does not belong to Slξ, meets the interior of &{S[t U { s}), absurdagain.

(β) Now, prove that

M(£\JSΪ.

Suppose that M e U* S{. Then

whence

impossible. Hence there exists x e M — (J;

572 TUDOR ZAMFIRESCU

(a) If x is an interior point of a simplex S'f with 5 ^ j <£ 8,then, following (a), M does not intersect the interior of any subsimplexS)f(l ^ 5) of

Hence, if Sίί c SU-j, then

n Sίί c J5f { U SI: S\ c Sjί} = {U S;6: S;6 c

whence Si2(M) ;z3 SI*, absurd.(b) If x is an interior point of the simplex Slk then, for

xreSlk, &(x,xr) intersects the interior of ί£(S\k U {xr}), absurd.Combining (a) and (β) we obtain that M(t\JiS\ and Si{M)a\Ji Si

contradicting one other. Therefore any Ma y such that S^2(M) — Cdoes not exist. Also, £ik(M) Φ C for k ^ 3, because £>i\s<ik-\M)) Φ C.Hence ω2(C) = 1.

(3) Prove that ω3(C) = 2. Suppose there exist a subset I c T "and a number & 3 such that S^k(M) = C Then M must containthe vertices xlf , x3δ of S. But

=) ?3(M) 3 £%\xu , α;35) -

whence Sik(M) Φ C, absurd. Because

ω,(C) = 2.(4) Prove that Ω(C) = 2 and J(C) = 3. If C would be 4-simplicial

convex, then 2α>4(C) 1, following Theorem 2, which is not possible.If C is i-simplicial convex, with I ^ 5, then suppose that M and k ^ 2satisfy 5>f(M) - C. Then

^f2(ikf) => ^(a?!, , x35) = U SU ,

absurd. Hence ω,(C) = 1.Thus β(C) = ω3(C) = 2 and Δ(C) = 3. The proof is complete.

5. Concluding remarks. Most of the results of [2] and of thepresent paper are purely algebraic and hence valid in vector spacesover arbitrary ordered fields.

Besides the operation f, the function ωι whose study was begunin § 2 of [2] and continued here is of special interest in the investiga-tion of ί-simplicial convexity. In §§ 3-4 of [2] we established someelementary facts concerning degree, order, and power. In order toinitiate a systematic study of simplicial convexity more informationabout these three functions should be obtained.

ON Z-SIMPLICIAL CONVEXITY IN VECTOR SPACES 573

REFERENCES

1. W. Bonnice and V. L. Klee, The generation of convex hulls, Math. Annalen 152(1963), 1-29.2. T. Zamίirescu, Simplicial convexity in vector spaces, Bull. Math. Soc. Sci. Math.R.S.R. 9 (1965), 137-149.

Received March 7, 1966.

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algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387P. H. Doyle, III and John Gilbert Hocking, A generalization of the Wilder

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athematics

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