Pairs of Compact Convex Sets - Fakultet strojarstva i ...titan.fsb.hr/~jmicic/Convexity and Applications... · Pairs of Compact Convex Sets, DCH-Functions and Semigroups 2 Minimality

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




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





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).






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) }.






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.






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) }.






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}






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






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)

}

.






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:






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)

.






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.






→ x1↑ ascent direction

← zero-level direction

↑ x2

←− descent direction

B

A






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






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






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)






Calculus:

Pinsker Formula:

Let X be a topological vector space and A,B ,C ∈ K(X ).Then:

(A + C ) ∨ (B + C ) = C + (A ∨ B).






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).






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:






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).






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






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 .





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.






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

.






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 .






-

x26

x1

A

-

x26

x1

B

-

x26

x1C

-

x26

x1

ւ D

-

x26

x1← separating hyperplane

A ∩ B






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.






{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)}






More suggestive: In terms of reducing fractions this looks like so:






Minimality

Geometric Approach

Pairs in general position:

-

x26

x1

A ∼=

B

-

x26

x1

·A′ւ B ′

The pair (A,B) is not minimal.






Minimality

In the above position the pair (A,B) is minimal

x2

-

6

x1

A

B






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).






Shape of a Convex Sets

SS

SS

SSo

Functional of a shape

•

•

•

•

•

•

��

��

�=

Functional of a shape

Functional of a shapeZ

ZZ

ZZ

ZZZ~

A






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.






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.






Polar Polytopes

Star of David

A

B

A + B






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






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.






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:






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.






Example

Orthogonal Lenses

A

B

A + B






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






Example

Turned Lenses

A

B

C

D

A + B = C + D (A,D) ∼ (C ,B)both pairs are not minimal






Pairs with piecewise smooth boundaries

Example

� x1

� x2

A

B

K = A + B = A − B

C







x1

x2

A

B

T = C + y

C

y

cutting hyperplane

K = A + B = A − B







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.





Translation PropertyInvariants

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.






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”.






Durer’s Melencolia I:






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:







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)







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






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.







-

x26

x1

(1,1)

E0

-

x26

x1

(1,1)

F0

-

x26

x1

(1,1)

U0

-

x26

x1

(1,1)

V0







-

x26

x1

(1,1)

(3,0)

E2

-

x26

x1

(3,1)

F2

-

x26

x1

(1,1)

(3,0)

U2

-

x26

x1

(3,1)

V2







6

x

z

y

-A0

z

x

y

6

-B0







z

x

y

6

-A1

z

x

y

6

-B1







6

-

A0 + B1 = B0 + A1

z

x

y






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)






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)





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






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 .






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






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.






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 .






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).






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




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.






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.






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.






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).






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.





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 .






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.






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.






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.






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:






front view back view

D

CD

C






B

A


B

A







D

C

DC






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.






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 ?






Thank you!


Documents

Pairs of Compact Convex Sets - Fakultet strojarstva i ...titan.fsb.hr/~jmicic/Convexity and Applications... · Pairs of Compact Convex Sets, DCH-Functions and Semigroups 2 Minimality