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Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Topics
1 Basic PropertiesPairs of Compact Convex Sets, DCH-Functions and Semigroups
2 MinimalityMinimality and Reduction
3 Uniqueness of Minimal PairsTranslation PropertyInvariants
4 Minimality under ConstraintsConvex Pairs and the Separation Property
5 Related ConceptsReduced PairsPairs of Bounded Closed Convex Sets
6 ApplicationsVirtual Polytopes and Piecewise Linear FunctionsData ClassificationCrystals
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Pairs of Compact Convex Sets
Notations
For a (real) topological vector space X let
K(X ) = {A ⊂ X | A non-empty compact convex }
be the set of all non-empty compact convex subsets of X and
K2(X ) = K(X )×K(X ).
For A,B ∈ K(X ) we put:
A+ B = {a + b | a ∈ A, b ∈ B} (Minkowski-Sum).
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
For pairs of compact convex sets
(A,B), (C ,D) ∈ K2(X )
we put(A,B) ∼ (C ,D)⇐⇒ A+ D = B + C
and(A,B) ¬ (C ,D)⇐⇒ A ⊆ C , B ⊆ D.
and[A,B ] = {(C ,D) ∈ K2(X ) | (C ,D) ∼ (A,B) }.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
For f ∈ X ∗ and K ∈ K(X ) we denote by
Hf (K ) = {z ∈ K | f (z) = maxy∈Kf (y)}
the (maximal) face of K with respect to f .
For A ∈ K(X ) we denote by E(A) the set of extreme points of Aand by
E0(A) = {x ∈ A | ∃f ∈ X ∗ \ {0} with Hf (A) = {x0} }
the set of exposed points.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
DCH-Functions
Pairs of Compact Convex Sets and DCH-Functions
The support function for A ∈ K(X ) is: PA : X ∗ −→ RwithPA(u) = sup
x∈A〈x , u〉.
and to (A,B) ∈ K2(X ) corresponds the difference of its supportfunctions, i.e.
ϕ(u) = PA(u)− PB(u),
which is also called the dual representation of the class
[A,B ] = {(C ,D) ∈ K2(X ) | (C ,D) ∼ (A,B) }.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
The one-to-one correspondence:
Pairs of compact convex sets The space of DCH-functions
K2(X ) = K(X )×K(X ) ⇐⇒ D(X ∗)
(A,B) ←→ PA − PB
(A,B) ∼ (C ,D) ←→ PA − PB = PC − PDm m
A+ D = B + C PA + PD = PB + PC
A ∨ B = conv(A ∪ B
)←→ max{PA,PB}
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Example
Let
ϕ : R2 −→ R with ϕ(x1, x2) = max{|x1|, |x2|} −
(1
2|x1|+ |x2|
)
.
–1–0.8
–0.6–0.4
–0.20
0.20.4
0.60.8
1
x
–1–0.8
–0.6–0.4
–0.20
0.20.4
0.60.8
1
y
–0.4
–0.2
0
0.2
0.4
z
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Thenϕ = PA − PB
with
A = conv{(1, 0), (0, 1), (−1, 0), (0,−1)}
and
B = conv
{
(1
2, 1), (−
1
2, 1), (−
1
2,−1), (
1
2,−1)
}
.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Ascent and Descent Directions:
Forϕ = PA − PB
withA = conv{(1, 0), (0, 1), (−1, 0), (0,−1)}
and
B = conv
{
(1
2, 1), (−
1
2, 1), (−
1
2,−1), (
1
2,−1)
}
one has:
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Steepest Ascent and Descent Directions
For a finite-dimensional spaces X = Rn, equipped with the Euclidean norm‖x‖ =
√〈x , x〉, the steepest ascent directions of ϕ = pA − pB ∈ D(X ) at the
point 0 ∈ Rn are the vectors
Desc(ϕ) =
{
x0 ∈ X | ‖x0‖ = 1 and ϕ(x0) = infx∈X‖x‖=1
ϕ(x)
}
and the steepest ascent directions are the vectors
Asc(ϕ) =
x0 ∈ X | ‖x0‖ = 1 and ϕ(x0) = sup
x∈X‖x‖=1
ϕ(x)
.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Steepest Ascent and Descent Directions
Theorem
Let X = Rn be equipped with the Euclidean norm and ϕ = pA − pB ∈ D(X ).Then
i) x0 ∈ Desc(ϕ) if and only if x0 = − w0 + v0‖w0 + v0‖
with
‖w0 + v0‖ = supw∈−B
infv∈A‖w + v‖.
ii) x0 ∈ Asc(ϕ) if and only if x0 =w0 + v0‖w0 + v0‖
with
‖w0 + v0‖ = supv∈−A
infw∈B‖w + v‖,
where w0 and v0 are the solutions of the above optimization problems.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
→ x1↑ ascent direction
← zero-level direction
↑ x2
←− descent direction
B
A
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
History and Semigroups
The Development of the Theory:
1911 Hermann Minkowski
1952 Hans Radstrom
1954 Lars V. Hormander
1966 Alexander.G. Pinsker
1968 Alexander M. Rubinov
1984 Vladimir F. Demyanov and Alexander M. Rubinov
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Hermann Minkowski
born 1864 in Aleksoty, a small village on thewestern side of the river Neman, which belonged to Guberia Augustowska and
was a part of the Kingdom of Poland, †1909 in Gottingen
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Calculus:
Order Cancellation Law:
Let X be a topological vector space, and A,B ,C ∈ K(X ).Then:
A+ C ⊆ B + C =⇒ A ⊆ B . (ocl)
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Calculus:
Pinsker Formula:
Let X be a topological vector space and A,B ,C ∈ K(X ).Then:
(A + C ) ∨ (B + C ) = C + (A ∨ B).
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Calculus:
Addition of Faces:
Let X be a locally convex vector space, f ∈ X ∗ andA,B ∈ K(X ). Then:
Hf (A + B) = Hf (A) + Hf (B).
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
The Minkowski Duality
Let X be a locally convex vector space and
D(X ) = {ϕ = pA − pB | (A,B) ∈ K2(X ∗)}
be the vector space of the differences of support functions.
Let (
K2(X ∗)/∼
,�
)
be the Minkowski-Radstrom-Hormander lattice of equivalence classes ofpairs of compact convex sets.
The Minkowski duality states:
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
The Minkowski Duality
The isomorphism:K2(X ∗)/
∼
−→ D(X )
withK2(X ∗)/
∼
∋ [A,B] 7→ ϕ = pA − pB ∈ D(X ),
which is order preserving, i.e.
pA − pB ¬ pC − pD if and only if [A,B]�[C ,D]
i.e.,A+ D ⊆ B + C .
Moreover
max{pA − pB , pC − pD} = p[(A+D)∨(C+B)] − p(B+D).
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
Semigroups
From the algebraic viewpoint of Hermann Minkowski about convexsets, we state that for a topological vector space X the space
(K(X ), ⋆,∨)
is an ordered commutative semigroup with cancellation propertywith “multiplication ⋆ ”
A ⋆ B = A + B Minkowski sum.
The Pinsker-Formula
(A + C ) ∨ (B + C ) = C + (A ∨ B).
gives the distributivity law for maximum and multiplication
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Pairs of Compact Convex Sets, DCH-Functions and Semigroups
and pairs of compact convex sets are fractions.
(A,B), (C ,D) ∈ K2(X )
and(A,B) ∼ (C ,D)⇐⇒ A+ D = B + C .
Analogon:
a
b,c
d∈ IQ and
a
b=c
d⇐⇒ ad = bc .
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Minimality and Reduction
Definition
Let X be a topological vector space. For pairs of compact convexsets (A,B), (C ,D) ∈ K2(X ).
We call a pair (A,B) ∈ K2(X ) minimal if it is minimal in the class[A,B ], i.e., if for any pair (C ,D) ∈ [A,B ] the relation(C ,D) ¬ (A,B) implies that (C ,D) = (A,B).
Theorem
Let X be a topological vector space. Then for any pair(A,B) ∈ K2(X ) there exists a pair (C ,D) ∈ [A,B ] which isminimal.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Reduction Method
Example
Reduction of a DCH-function:
Considerh : R2 → R
with
h(x1, x2) = max{0, x1, x2, x1 + x2}︸ ︷︷ ︸
PA
−max{x1, x2, x1 + x2}︸ ︷︷ ︸
PB
= max{0, x1, x2}︸ ︷︷ ︸
PC
−max{x1, x2}︸ ︷︷ ︸
PD
.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Reduction by cutting
Example
Hence the pairs (A,B) and (C ,D) are equivalent
-
x26
x1
A
-
x26
x1
B
-
x26
x1C
-
x26
x1
ւD
but have no common summand. Moreover C ⊂ A and D ⊂ B .
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
-
x26
x1
A
-
x26
x1
B
-
x26
x1C
-
x26
x1
ւ D
-
x26
x1← separating hyperplane
A ∩ B
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
General Result
Reduction Technique
All parts of two compact convex sets which can be translatedonto each other can be cut off without leaving the equivalenceclass.
CutA+f2,z2+d2
and B+f2,z2
andA−f1,z1+d1 and B−f1,z1.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
{x ∈ X | f2(x) = f2(z2 + d2)}
A
A+f2,z2+d2
A−f1,z1+d1
{x ∈ X | f2(x) = f2(z2)}
B
The general case
B−f1,z1
B+f2,z2
{x ∈ X | f1(x) = f1(z1 + d1)}
{x ∈ X | f1(x) = f1(z1)}
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
More suggestive: In terms of reducing fractions this looks like so:
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Minimality
Geometric Approach
Pairs in general position:
-
x26
x1
A ∼=
B
-
x26
x1
·A′ւ B ′
The pair (A,B) is not minimal.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Minimality
In the above position the pair (A,B) is minimal
x2
-
6
x1
A
B
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Shape of a Convex Sets
Let X be a locally convex vector space and A ∈ K(X ).A subset S ⊆ X ∗ \ {0} with
conv(⋃
f ∈S
Hf (A)) = A
is called a shape of A. We write S(A) for S.
We put:Sp(A) = {f ∈ S(A) | card(Hf (A)) = 1}
andSl (A) = S(A) \ Sp(A).
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Shape of a Convex Sets
SS
SS
SSo
Functional of a shape
•
•
•
•
•
•
��
��
�=
Functional of a shape
Functional of a shapeZ
ZZ
ZZ
ZZZ~
A
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Geometric Citerium
Theorem
Let X a real locally convex vector space and let A,B ⊂ X benon-empty compact convex sets. Let us assume that there is ashape S(A) of A satisfying the following conditions:
1 for every f ∈ S(A) , card(Hf (B)) = 1
2 for every f ∈ Sl(A) and every b ∈ B , the conditionSl(A) + (b − Hf (B)) ⊆ A implies b = Hf (B).
3 for every f ∈ Sp(A) , Hf (A)− Hf (B) ∈ E(A − B)
or conversely by interchanging A and B .
Then the pair (A,B) ∈ K 2(X ) is minimal.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Polar Polytopes
If X = Rn and P ∈ K(X ) be a polytope then the polar polytope isdefined by:
Po = {u ∈ Rn | supx∈P〈u, x〉 ¬ 1},
where 〈., .〉 denotes the inner product of Rn.
Proposition:For every polytope P ∈ K(Rn) the pair
(P ,Po) ∈ K2(Rn)
is minimal.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
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Polar Polytopes
Star of David
A
B
A + B
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Dual of the Star of David:
–1–0.8
–0.6–0.4
–0.20
0.20.4
0.60.8
1
x
–1–0.8
–0.6–0.4
–0.20
0.20.4
0.60.8
1
y
–0.4
–0.2
0
0.2
0.4
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Minimal Pairs
Algebraic Criterium
Assume that the pair (A,B) ∈ K2(X ) is not minimal. Then thereexists an equivalent pair (A′,B ′) ∈ K2(X ) with A′ ⊆ A andB ′ ⊆ B where at least one inclusion is proper. Now from
A+ B ′ = B + A′ ⊂ A+ B
it follows that there exists a proper compact convex subsetK ⊂ A+ B namely K = A′ + B = B ′ + A for which A and B aresummands.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
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Minimality and Reduction
Minimal Pairs
Hence:
Theorem
A pair (A,B) ∈ K2(X ) is minimal if and only if, there exists noproper compact convex subset K ⊂ A+ B such that A and B is asummand of K .
Using this characterization, we show:
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Minimal Pairs
Theorem
Let X be a real Banach space, and let(A,B) ∈ K2(X ).If for every exposed point a + b ∈ E0(A+ B) with a ∈ E0(A),b ∈ E0(B) there exists b1 ∈ E0(B) or a1 ∈ E0(A) such thata + b1 ∈ E0(A+ B) and a− b1 ∈ E(A− B)ora1 + b ∈ E0(A+ B) and a1 − b ∈ E(A− B).
Then the pair (A,B) is minimal.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Example
Orthogonal Lenses
A
B
A + B
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Dual of the Orthogonal Lenses:
–2
–1
0
1
2
x
–2
–1
0
1
2
y
–0.8–0.6–0.4–0.2
00.20.40.60.8
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Example
Turned Lenses
A
B
C
D
A + B = C + D (A,D) ∼ (C ,B)both pairs are not minimal
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Related ConceptsApplications
Minimality and Reduction
Pairs with piecewise smooth boundaries
Example
� x1
� x2
A
B
K = A + B = A − B
C
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Pairs with piecewise smooth boundaries
x1
x2
A
B
T = C + y
C
y
cutting hyperplane
K = A + B = A − B
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Minimality and Reduction
Pairs with piecewise smooth boundaries
Defintion
A set A ∈ K(Rn) is called locally ǫ−smooth in x0 ∈ ∂A if thereexists a neighborhood U of x0 such that for every x ∈ A ∩ U thereexists y ∈ Rn with x ∈ y + ǫBn ⊂ A, where B
n = B(0, 1) is theclosed Euclidean unit ball in Rn.
Theorem
Let A,B ∈ K(Rn) and let A be locally ǫ−smooth in x0 ∈ ∂A andB be be locally ǫ−smooth in y0 ∈ ∂B . If there exists a linearfunctional f ∈ (Rn)∗ with Hf A = {x0} and Hf B = {y0}, then thepair (A,B) is not minimal.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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General Results
Theorem
Let (A,B), (C ,D) be equivalent minimal pairs in the plane R2.Then there exists a vector x ∈ R2 such that C = A+ x andD = B + x .
In the 3-dimensional space exist already equivalent minimal pairswhich are not related by a translation.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Examples
A+ x
B − x
A+ B
This figure shows the Minkowski sum of the first known example of a minimal pair not
having the translation property. It is similar to the polyhedron from Albrecht Durer’s
“Melencolia I”.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Durer’s Melencolia I:
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
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Translation PropertyInvariants
Uniqueness of Minimal Pairs
Theorem
For a topological vector space X let (A1,B1), (A2,B2) ∈ K2(X ) be
two equivalent minimal pairs which are not related by translation.
Then there exists a non-countable family (Aλ,Bλ), λ ∈ Λ ofmimimal pairs that are all equivalent to (A1,B1) and no (Aλ,Bλ)is a translate of (Aµ,Bµ) for λ 6= µ.
An explicit construction of such a family goes as follows:
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Uniqueness of Minimal Pairs
The General Frustum
Let X be a locally convex vector space, f ∈ X ∗ and z ∈ X with f (z) 6= 0.Moreover let E ,F ∈ K(X ) be such that E ,F ⊂ f −1(0). Then the set
A = E ∨ (F + {z})
is called a Frustum over E and F .
F + z
Ef−1(0)
f−1(α)
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Uniqueness of Minimal Pairs
Theorem
Let X be a locally convex vector space, f ∈ X ∗ a continuous linearfunctional, z ∈ X , with f (z) 6= 0 and for i ∈ {0, 1} letEi ,Fi ,Ui ,Vi ∈ K(X ) be non-empty compact convex sets, withEi ,Fi ,Ui ,Vi ⊂ f
−1(0). Let Ai = IF (Ei ,Fi ) = Ei ∨ (Fi + {z}) andBi = IF (Ui ,Vi ) = Ui ∨ (Vi + {z}) be general frusta.Then
(A0,B0) ∼ (A1,B1)
if and only if
i) (E0,U0) ∼ (E1,U1)
ii) (E0 + V1) ∨ (F0 + U1) = (E1 + V0) ∨ (F1 + U0)
iii) (F0,V0) ∼ (F1,V1)
%Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Uniqueness of Minimal Pairs
Theorem
Let f ∈ (R3)∗ be given by f (x) = f ((x1, x2, x3)) = x3 and putz = e3 = (0, 0, 1) ∈ R3. Now for α 0 define the following sets:
i) Eα = conv{(0, 0, 0), (1, 1, 0), (1+ α, 0, 0)}ii) Fα = conv{(0, 1, 0), (α, 0, 0), (1+ α, 1, 0)}iii) Uα = conv{(0, 0, 0), (0, 1, 0), (1, 1, 0), (1+ α, 0, 0)}iv) Vα = conv{(0, 1, 0), (α, 0, 0), (1+ α, 0, 0), (1+ α, 1, 0)}.
Then the families of general frusta
Aα = IF (Eα,Fα) = Eα ∨ (Fα + {z})
Bα = IF (Uα,Vα) = Uα ∨ (Vα + {z})
form a family of equivalent minimal pairs (Aα,Bα) ∈ K2(R3) which are notconnected by translations.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Uniqueness of Minimal Pairs
-
x26
x1
(1,1)
E0
-
x26
x1
(1,1)
F0
-
x26
x1
(1,1)
U0
-
x26
x1
(1,1)
V0
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Translation PropertyInvariants
Uniqueness of Minimal Pairs
-
x26
x1
(1,1)
(3,0)
E2
-
x26
x1
(3,1)
F2
-
x26
x1
(1,1)
(3,0)
U2
-
x26
x1
(3,1)
V2
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Uniqueness of Minimal Pairs
6
x
z
y
-A0
z
x
y
6
-B0
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
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Translation PropertyInvariants
Uniqueness of Minimal Pairs
z
x
y
6
-A1
z
x
y
6
-B1
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Translation PropertyInvariants
Uniqueness of Minimal Pairs
6
-
A0 + B1 = B0 + A1
z
x
y
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
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Translation PropertyInvariants
The Invariance of the Affine Dimension
Let X be a locally convex vector space and C ∈ K(X ) be a nonemptycompact convex subset. Then for every y ∈ C the set
Cy = span(C−y) = cl({z ∈ X | z =
n∑
i=1
λi (ci−y), c1, ..., cn ∈ C , n ∈ N})
is the smallest closed linear subspace containing C − y or equivalentlythe intersection of all closed linear subspace containing C − y .
The affine dimension and codimension is defined by:
dim aff(C ) = dim(Cy )
andcodim aff(C ) = codim(Cy ) = dim(X/
Cy)
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Translation PropertyInvariants
The Invariance of the Affine Dimension
The following statement holds:
Theorem
Let X be a locally convex vector space and and(A,B), (C ,D) ∈ K2(X ) be equivalent minimal pairs.
Thendim aff(A ∪ B) = dim aff(C ∪D)
andcodim aff(A ∪ B) = codim aff(C ∪ D)
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Convex Pairs and the Separation Property
Convex Pairs
Definition
A pair (A,B) ∈ K2(X ) is called convex if A ∪ B is a convex set.
A B
A ∩B
A
B
A ∩B
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Convex Pairs and the Separation Property
Convex Pairs
For convex pairs holds:
Theorem
The following statements are equivalent:
i) The pair (A,B) ∈ K2(X ) is convex.
ii) The set A ∩ B separates the sets A and B , i.e.for every a ∈ A and b ∈ B the line segment
between a and b intersects A ∩ B.
iii) The following formula holds:
A+ B = A ∪ B + A ∩ B .
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Convex Pairs and the Separation Property
Separation of Sets by Sets
A convex set S ∈ K(X ) separates A,B ∈ K(X ) if for every a ∈ Aand b ∈ B the line segment between a and b intersects S .
A
BS
a
b
x
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Convex Pairs and the Separation Property
Separation Property
Theorem
The set S ∈ K(X ) separates A,B ∈ K(X ) if and only if
A+ B ⊆ A ∨ B + S (sl)
Theorem
In a topological vector space the order cancellation law (ocl) andthe separation law (sl) are equivalent.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
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Convex Pairs and the Separation Property
Conditional Minimality
Definition
Let X be a topological vector space.
A convex pair (A,B) ∈ K2(X ) is called minimal convex if andonly if for every equivalent convex pair (C ,D) ∈ K2(X ) therelation (C ,D) ¬ (A,B) implies C = A and B = D.
For a given C ∈ K(X ) a pair (A,B) ∈ K2(X ) is calledC-minimal if the pair (A+ C ,B + C ) is convex, and if forevery C1 ∈ K(X ) with C1 ⊆ C and such that(A+ C1,B + C1) is a convex pair it follows that C1 = C .
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
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Convex Pairs and the Separation Property
Conditional Minimality
Characterizations
Theorem
Let X be a topological vector space. Then the convex pair(A,B) ∈ K2(X ) is minimal convex if and only if the pair(A ∩ B ,A ∪ B) is minimal.
Theorem
Let X be topological vector space and C ∈ K(X ).Then the pair (A,B) ∈ K2(X ) is C-minimal if and only if thereexists a D ∈ K(X ) such that the pair (C ,D) is minimal andequivalent to (A ∨ B ,A+ B).
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
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Convex Pairs and the Separation Property
Minimal and Minimal Convex Pairs
In R2 put A = {(0, 1)} ∨ {( 12
√3,− 1
2)} ∨ {(− 1
2
√3,− 1
2)} and B = − A.
B
A
A + B
The pairs (A,B) and (A ∨ B,A+ B) are minimal and (A+ A ∨ B,B + A ∨ B)
is minimal convex.Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
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Reduced PairsPairs of Bounded Closed Convex Sets
Definition and some Properties
The notion of a reduced pair has been introduced in 1996 by Chr.Bauer and R. Schneider.
Definition
Let X be topological vector space.Then a pair (A,B) ∈ K2(X ) iscalled reduced if and only if[A,B ] = {(A + C ,B + C ) | C ∈ K(X )}.
All reduced pairs of sets are minimal, but not all minimal pairs arereduced.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Reduced PairsPairs of Bounded Closed Convex Sets
Definition and some Properties
Reduced Pairs of Polytopes
Chr. Bauer proved the following characterization of reduced pairs of polytopes.
Definition
Let A and B be two polytopes in Rn. We call an edge (one-dimensional face)k of A and an edge l of B equiparallel if k = Hf (A) and l = Hf (B) for somelinear functional f ∈ (Rn)∗.
Now the following characterization holds:
Theorem
A pair (A,B) ∈ K2(Rn) of polytopes is reduced if and only if A and B have noequiparallel edges.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Reduced PairsPairs of Bounded Closed Convex Sets
Notations
The Minkowski-Radstrom-Hormander lattice
Let X a topological vector space and
B(X ) = {A | A 6= bounded closed subset of X}.
Put
A+ B = A+ B ,
then (B(X ), + ) is a semigroup which satisfies the cancellation law.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Reduced PairsPairs of Bounded Closed Convex Sets
Notations
For (A,B), (C ,D) ∈ B2(X ), define
(A,B) ∼ (C ,D) if and only if A+ D = B+ C .
The relation “ ∼” is a relation of equivalence and [A,B] is a classequivalence of the pair (A,B).
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
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Reduced PairsPairs of Bounded Closed Convex Sets
Minimal Pairs of Bounded Closed Convex Sets
Theorem
Let X be a reflexive topological vector space. Then every class[A,B ] ∈ B2(X )/ ∼ contains a minimal element (A0,B0) .
Theorem
Let X = c0, c , or l∞. Then there exists a class [A,B ] ∈ B2(X )/ ∼
that contains no minimal elements.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
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Virtual Polytopes and Piecewise Linear FunctionsCrystals
Continuous Selections
Definition
Let U ⊆ Rn be an open subset and f1, ..., fm : U −→ R be continuous functions.
A continuous function f : U −→ R is called a continuous selection of thefunctions f1, ..., fm if for every x ∈ U the set
I (x) = {i ∈ {1, ...,m} | fi (x) = f (x)}
is nonempty.
We denote by CS(f1, ..., fm) the set of all continuous selections of f1, ..., fm andthe set I (x) is called the active index set of f at the point x .
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Continuous Selections
A continuous selection of a certain type of functions, for instancedifferentiable, linear or affine is called a piecewise differentiable,linear or affine function.
Typical examples of continuous selections are the functions
fmax = max(f1, ..., fm), fmin = min(f1, ..., fm)
or, more generally, any finite superposition of maximum andminimum operations over subsets of the functions f1, ..., fm.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Virtual Polytopes
Definition
A pair (A,B) ∈ K2(Rn) is called a virtual polytope.
Note that for every virtual polytope (A,B) ∈ K2(Rn) the dual representation ofthe class [A,B] given by:
ϕ(u) = PA(u)− PB(u)
is a piecewise linear function.
Now we consider a very special type of piecewise linear functions which appears
in the local representation of a piecewise smooth function around a non
degenerated critical point as the nonsmooth part in the second Morse Lemma.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Virtual Polytopes
We determine the minimal pairs of compact convex sets which correspond tothe continuous selections of four linear functions
li : R3 −→ R
with
li (x) = xi for i ∈ {1, 2, 3} and l4(x) = −3∑
i=1
xi
in R3.
The set CS(x1, x2, x3,−∑3i=1xi ) in R
3 consists of 166 continuous selectionsbut only by 16 essential different minimal pairs of polytopes. Three out of 16cases are minimal pairs that are not unique minimal representations in theirown quotient classes.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
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Virtual Polytopes
Theorem
The set
CS(x1, x2, x3,−3∑
i=1
xi )
consists of 166 continuous selections which are represented by 16 essentialdifferent minimal pairs. Three out of these 16 cases are minimal pairs that arenot unique minimal representations in their own quotient classes.
We show this three pairs:
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
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Virtual Polytopes and Piecewise Linear FunctionsCrystals
front view back view
D
CD
C
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
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Virtual Polytopes and Piecewise Linear FunctionsCrystals
B
A
front view back view
B
A
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Virtual Polytopes and Piecewise Linear FunctionsCrystals
front view back view
D
C
DC
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Virtual Polytopes and Piecewise Linear FunctionsCrystals
Data ClassificationThere exist interesting applications of the separation law indata classification. In the case of medical data it often happensthat the set of data which can not be uniquely assigned to adata type is quite large.
In such cases the usual classification methods by usingseparating hyperplanes fails and the separation law is used fordetermining constraints for an optimal separation of the sets.
CrystalsStructural analysis of crystals —- Crystal growth formula.
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Virtual Polytopes and Piecewise Linear FunctionsCrystals
Open Questions
Are minimal pairs generated by pairs of polytops also polytops ?
Does in every non-reflexive locally convex topological vector space, thereexist an equivalence class [A,B] ∈ B2(X )/ ∼ containing no minimalelements ?
If (A,B), (C ,D) are equivalent minimal pairs, then
dim aff(A ∪ B) = dim aff(C ∪ D)
andcodim aff(A ∪ B) = codim aff(C ∪ D)
is only one known invariant of sets minimal pairs: which are others ?
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets
Basic PropertiesMinimality
Uniqueness of Minimal PairsMinimality under Constraints
Related ConceptsApplications
Virtual Polytopes and Piecewise Linear FunctionsCrystals
Thank you!
Jerzy Grzybowski, Diethard Pallaschke and Ryszard Urbański Pairs of Compact Convex Sets