Minimal pairs of convex sets which share a recession cone
aa r X i v : . [ m a t h . F A ] D ec MINIMAL PAIRS OF CONVEX SETS WHICH SHAREA RECESSION CONE
JERZY GRZYBOWSKI AND RYSZARD URBA ´NSKI
Abstract.
Robinson introduced a quotient space of pairs of con-vex sets which share their recession cone. In this paper minimalpairs of unbounded convex sets, i.e. minimal representations ofelements of Robinson’s spaces are investigated. The fact that aminimal pair having property of translation is reduced is proved.In the case of pairs of two-dimensional sets a formula for an equiv-alent minimal pair is given, a criterion of minimality of a pair ofsets is presented and reducibility of all minimal pairs is proved.Shephard–Weil–Schneider’s criterion for polytopal summand of acompact convex set is generalized to unbounded convex sets. Anapplication of minimal pairs of unbounded convex sets to Hart-man’s minimal representation of dc-functions is shown. Examplesof minimal pairs of three-dimensional sets are given.
1. Introduction
For a family C ( R n ) of all nonempty closed convex subsets of R n theaddition A + B := { a + b | a ∈ A, b ∈ B } is called a Minkowski or vector or algebraic sum of these sets. For A, B ∈ C ( R n ) the modifiedaddition A ˙+ B := cl ( A + B ) turns the family C ( R n ) into a commutativesemigroup with a neutral element { } . Moreover, for all A, B ∈ C ( R n )and all s, t > s ( tA ) = s ( tA ), t ( A ˙+ B ) = tA ˙+ tB , ( s + t ) A = sA ˙+ tA , 1 A = A , and 0 A = { } . A relation ( A, B ) ∼ ( C, D ) : ⇐⇒ A ˙+ D = B ˙+ C is not transitive because in C ( R n ) a cancellation law A ˙+ B = B ˙+ C = ⇒ A = C does not hold true. Therefore, the family C ( R n ) cannot be embedded into a vector space.However, the family B ( R n ) of all nonempty closed bounded convexsubsets of R n can be embedded into a vector space, see Minkowski[24]. In a case of infinitely dimensional topological vector spaces asemigroup of nonempty closed bounded convex sets can be embed-ded into Minkowski–R˚adstr¨om–H¨ormander space, see R˚adstr¨om [29],H¨ormander [23], Drewnowski [11] and Urba´nski [36]. Mathematics Subject Classification.
Key words and phrases.
Minkowski addition, recession cone, Minkowski–R˚adstr¨om–H¨ormander spaces, minimal pairs of convex sets, dc-functions.
Quotient classes of pairs of convex sets are elements of Minkowski–R˚adstr¨om–H¨ormander spaces. Sets in a given class can be arbitrar-ily large. The best representation of such a class would be inclusion-minimal pair. Inclusion-minimal pairs were studied by Bauer [5], Scholtes[26, 34], Pallaschke [15, 16, 27, 28] and by the authors [12, 13, 19, 20]in connection with quasidifferential calculus. Quasidifferential calculuswas developed by Demyanov and Rubinov [8] and studied by many au-thors including Zhang, Xia, Gao and Wang [38] Basaeva, Kusraev andKutateladze [4], Antczak [2], Abbasov [1], Dolgopolik [10] and others.MRH spaces and basic facts about minimal pairs of convex sets arepresented in Section 7. An embedding of a semigroup of convex sets isenabled by a cancellation law which was studied for its own sake by theauthors [20] and recently generalized to cornets by Moln´ar and P´ales[25].Robinson [30] proved an order cancellation law A + B ⊂ B + C = ⇒ A ⊂ C. (olc)for A, B, C from a family of unbounded closed convex sets C V ( R n )sharing a common recession cone V . Here, V is a closed convex cone in R n and a recession cone is defined as recc A := { x ∈ R n | x + A ⊂ A } .A family C V ( R n ) with Minkowski addition is a semigroup by Corollary9.1.1 in [31] and as such can be embedded into a vector space. Inthis family the closed convex cone V is a neutral element, A ˙+ B = A + B , and multiplication by 0 has to be modified by 0 A := V for A, B ∈ C V ( R n ). Since a cancellation law holds true, the relation ” ∼ ”is transitive. We put [ A, B ] := [(
A, B )] ∼ . Theorem 1.1. (Robinson, [30])
The family of quotient classes f R nV := C V ( R n ) / ∼ with with the addition [ A, B ] + [
C, D ] := [ A + C, B + D ] and the multiplication t [ A, B ] := (cid:26) [ tA, tB ] , t > − tB, − tA ] , t < is a smallestvector space into which the semigroup C V ( R n ) can be embedded. The embedding is defined by C V ( R n ) ∋ A [ A, V ] ∈ f R nV . In thevector space f R nV the neutral element is [ V, V ] and the opposite elementto [
A, B ] is − [ A, B ] = [
B, A ].If the cone V is trivial, i.e. V = { } the family C V ( R n ) coincides witha well studied family B ( R n ) of all nonempty compact convex sets, i.e.of convex bodies.Robinson’s theorem was generalized for closed convex sets in a Banachspace by Bielawski and Tabor [6]. INIMAL PAIRS OF CONVEX SETS WHICH SHARE A RECESSION CONE 3
Balashov and Polovinkin in their interesting paper [3] extended to un-bounded sets the notion of generating sets. In a similar manner thispaper extends to unbounded sets the notion of minimal pairs of sets.In Section 2 we present a definition and a theorem of existence ofminimal pairs of sets from C V ( R n ) and the property of translation ofminimal pairs. We give properties of a kernel of minimality B ∗ of apair ( A, B ), i.e. a set of all such points x that a pair ( A − x, B − x ) isminimal. We also give a number of examples.In Section 3 we prove that a minimal pair of sets is reduced if and onlyif it has the property of translation.Properties of minimal pairs of two-dimensional sets are studied in Sec-tion 4. We give a criterion for being a summand in Proposition 4.2, aformula for an equivalent minimal pair in Theorem 4.4, a criterion ofminimal pair in Theorem 4.5 and prove the reducibility of all minimalpairs in Theorem 4.6.We generalize Shephard–Weil–Schneider’s criterion, i.e. Th. 3.2.11 in[33], to polytopal summands of unbounded convex sets in Theorem 5.2.We also extend Bauer’s criterion [5] of reduced pairs of polytopes to V -polytopes in Theorem 5.5.In Section 6 we present an application of minimal pairs of unboundedconvex sets to a minimal, according to Hartman [22], representation ofdc-functions.We complete our paper with two appendices. In Section 7 we presentselected facts from [16] on minimal pairs of bounded convex sets used inour proofs. In Section 8 we present Minkowski duality between convexsets and sublinear functions needed in the proof of Theorem 5.2.
2. Minimal pairs of unbounded convex sets
Let V be a closed convex cone in R n and A, B ∈ C V ( R n ). A quotientclass [ A, B ] is ordered in the following way( A , B ) ≺ ( A , B ) ⇐⇒ A ⊂ A , B ⊂ B . If a recession cone V is not trivial then a pair ( A + v, B + v ) , v ∈ V \ { } is smaller than ( A, B ) hence no pair (
A, B ) ∈ C V ( R n ) is minimal.Therefore, we say that a pair ( A, B ) is 0- minimal if (
A, B ) is a minimalelement in a subset { ( C, D ) ∈ [ A, B ] | ∈ D } of a quotient class [ A, B ].The definition of 0-minimality seems very natural. In the case of asemigroup B ( R n ) = C { } ( R n ) of bounded closed convex sets the ex-istence of a minimal pair is guaranteed by the fact that a chain of JERZY GRZYBOWSKI AND RYSZARD URBA ´NSKI compact sets has a nonempty intersection. In the case of C V ( R n ) anintersection of a chain of sets containing 0 contains the cone V .Every quotient class [ A, B ] ∈ C V ( R n ) / ∼ contains a 0-minimal pair. Thefollowing theorem was proved by Grzybowski and Przybycie´n [18] inmuch more general, possibly infinite dimensional, case. Theorem 2.1 (existence of a -minimal pair). For every pair ( A, B ) ∈ C V ( R n ) with ∈ B there exists an equivalent -minimal pair ( A ′ , B ′ ) such that A ′ ⊂ A, B ′ ⊂ B . Unlike in the case of minimal pairs of compact convex sets a pair(
A, B ) ∈ C V ( R n ) may be 0-minimal and a translated pair ( A − x, B − x )may not. We call a set B ∗ := { x ∈ B | ( A − x, B − x ) is 0-minimal } a kernel of minimality of the pair ( A, B ). Obviously, B ∗ ⊂ B .By L V = V ∩ ( − V ) we denote the subspace of lineality of the cone V .Let us notice that for a pair ( A, B ) ∈ C V ( R n ) we have the followingequality { b ∈ B | ( A − b, B − b ) ≺ ( A, B ) } = B ∩ ( − V ) . ( ∗ )The following proposition holds true. Proposition 2.2.
Let ( A, B ) ∈ C V ( R n ) . If x ∈ B ∗ then B ∩ ( x − V ) = x + L V .Proof. Let b ∈ B ∩ ( − V ), then from ( ∗ ) we have ( A − b, B − b ) ≺ ( A, B ).Assume that 0 ∈ B ∗ . Then the pair ( A, B ) is 0-minimal and we get B − b = B . Hence b ∈ L V and L V ⊂ B ∩ ( − V ) ⊂ L V . If x ∈ B ∗ then( A − x, B − x ) is 0-minimal and ( B − x ) ∩ ( − V ) = L V . (cid:3) Proposition 2.2 says that the kernel of minimality is contained in thesubset of minimal elements of B with respect to the preorder V gen-erated by the cone V . Notice also that if the cone V is nontrivial thenthe set B ∗ is contained in the boundary of B . Lemma 2.3 ( B ∗ is an extreme subset of B ). Let ( A, B ) ∈ C V ( R n ) .If x, y ∈ B and ( x + y ) / ∈ B ∗ then x, y ∈ B ∗ .Proof. Denote z = ( x + y ) /
2. By Theorem 2.1 there exists a 0-minimalpair ( A ′ − x, B ′ − x ) ≺ ( A − x, B − x ). Hence z = ( x + y ) / ∈ ( B ′ + B ) / A ′ A − z, B ′ B − z ) ≺ ( A − z, B − z ) . Since the pair ( A − z, B − z ) is 0-minimal, we obtain B ′ / B/ B .By the cancellation law (olc) we get B ′ / B/
2, and B ′ = B . Thenthe pair ( A − x, B − x ) is 0-minimal, and x ∈ B ∗ . (cid:3) INIMAL PAIRS OF CONVEX SETS WHICH SHARE A RECESSION CONE 5
Corollary 2.4.
Let ( A, B ) ∈ C V ( R n ) . If E ⊂ B is a convex extremesubset of B and the relative interior of E intersects with B ∗ then E ⊂ B ∗ . Let A ∈ C ( R n ), u, u , ..., u k ∈ R n . Let A ( u ) be a support set de-fined by A ( u ) = { a ∈ A | h a, u i = max x ∈ A h x, u i} . Let A ( u , ..., u k ) = A ( u , ..., u k − )( u k ) be an iterated support set. Notice that any subsetof A is a convex extreme subset of A if and only if it is an iteratedsupport set of A . In particular a singleton consisting of an extremepoint of A is an extreme subset of A .If A, B, C, D ∈ C V ( R n ) and ¯ u = ( u , ..., u k ) ∈ ( R n ) k and V ( u ) = L V then the following significant facts hold true( A + B )(¯ u ) = A (¯ u ) + B (¯ u )and ( A, B ) ∼ ( C, D ) = ⇒ ( A (¯ u ) , B (¯ u )) ∼ ( C (¯ u ) , D (¯ u )) . The following proposition shows that kernels of minimality of pairs(
A, B ) and (
B, A ) ”lie on the same side”, respectively, of sets B and A . Proposition 2.5.
Let ( A, B ) ∈ C V ( R n ) , ¯ u = ( u , ..., u k ) ∈ ( R n ) k and V ( u ) = L V . If B (¯ u ) ⊂ B ∗ then A (¯ u ) ⊂ A ∗ where A ∗ = { x ∈ A | ( B − x, A − x ) is 0-minimal } .Proof. Let y ∈ A (¯ u ). Then by Theorem 2.1 there exists a 0-minimalpair ( B ′ − y, A ′ − y ) ≺ ( B − y, A − y ). Since y ∈ A ′ ⊂ A and y ∈ A (¯ u ), weobtain A ′ (¯ u ) ⊂ A (¯ u ). Since ( B ′ , A ′ ) ∼ ( B, A ), we get B ′ + A = A ′ + B ,and B ′ (¯ u ) + A (¯ u ) = A ′ (¯ u ) + B (¯ u ) ⊂ A (¯ u ) + B (¯ u ). Hence by the orderlaw of cancellation B ′ (¯ u ) ⊂ B (¯ u ). Consider any x ∈ B ′ (¯ u ). Since B (¯ u ) ⊂ B ∗ , the pair ( A − x, B − x ) is 0-minimal. Moreover, x ∈ B ′ and ( A ′ − x, B ′ − x ) ≺ ( A − x, B − x ). Then B ′ = B , and we have justproved that y ∈ A ∗ (cid:3) A pair (
A, B ) or a class [
A, B ] is said to have a property of translationof -minimal pairs if all equivalent 0-minimal pairs in [ A, B ] are con-nected by translation. This property of translation is distinct from aproperty of translation of minimal pairs of bounded sets. If the cone V is not trivial we write just ’property of translation’ because there isno possibility of misunderstanding.For ( A, B ) ∈ C { } ( R n ) a property of translation of 0-minimal pairsfollows from a property of translation of minimal pairs but not theother way around. All pairs of flat compact convex sets from C { } ( R )satisfy the property of translation of minimal pairs [5, 12, 34], but JERZY GRZYBOWSKI AND RYSZARD URBA ´NSKI
Example 2.10(i) presents a number of equivalent 0-minimal pairs notconnected by translation.
Proposition 2.6 (characterization of a kernel of -minimal pair). Let a -minimal pair ( A, B ) ∈ C V ( R n ) have the property of translation.Then the following assertions hold :(a) The set { ( A − x, B − x ) | x ∈ B ∗ } is a set of all -minimal pairsof the class [ A, B ] . (b) x ∈ B ∗ if and only if B ∩ ( x − V ) = x + L V . (c) B = B ∗ + V .Proof. (a) Let ( C, D ) ∈ [ A, B ] be a 0-minimal pair, then by a propertyof translation D = B − z for some z ∈ R n . Since 0 ∈ D we get z = x for a some x ∈ B .(b) Let ( B − x ) ∩ ( − V ) = L V , we have 0 ∈ B − x . By (a) thereexist a 0-minimal pair ( A − z, B − z ) such that B − z ⊂ B − x . Hence( B − x ) − ( z − x ) = B − z ⊂ B − x . Now, by ( ∗ ) applied to ( A − x, B − x )we get z − x ∈ ( B − x ) ∩ ( − V ) = L V . Hence B − z = B − x + V − ( z − x ) ⊃ B − x and we get B − z = B − x .(c) By (a) for any b ∈ B there exists x ∈ B ∗ such that B − x ⊂ B − b .Then B + b − x ⊂ B , and b − x ∈ V . Therefore, b = x +( b − x ) ⊂ B ∗ + V ,and we get B ∗ + V ⊂ B ⊂ B ∗ + V . (cid:3) Remark 2.7.
Let us notice that in case of a 0-minimal pair (
A, B )not having the property of translation the equality B = B ∗ + V mayhold true, see the pair ( b A , b B ) in Example 2.10(ii), or not, see the pair( b A , b B ) in Example 2.10(ii).Obviously, any pair ( A, V ) has property of translation. Moreover, itis a unique 0-minimal pair in a quotient class [
A, V ]. The followingexample gives all 0-minimal pairs in a quotient class [
V, B ]. Example 2.8.
Let n = 2, A = V = { } × R + be a ray and B = { ( x , x ) ∈ R | x > x } be an epigraph of a quadratic function. A pair( A, B ) is obviously 0-minimal. By Proposition 2.6 a pair ( A − x, B − x )is 0-minimal if and only if B ∩ (( x , x ) − V ) = { ( x , x ) } , where L V = { (0 , } . This equality holds true exactly when x = x , x ∈ R . Theset B ∗ is equal to the boundary of the set B . Notice that A ∗ = { (0 , } .In R there exist equivalent minimal pairs not connected by transla-tion. The following example was given as Example 4.1 in [18]. In thatexample a pair ( C, D ) – not showed here – was incorrectly presentedas 0-minimal.
Example 2.9.
Let V = { x ∈ R | x = x = 0 , x } , B = conv { ( − , − , , ( − , , − , (1 , , , (1 , − , − } + V , A = conv ( B ∪ { ( − , , − , (2 , , − } ) + V , INIMAL PAIRS OF CONVEX SETS WHICH SHARE A RECESSION CONE 7 F = conv { (0 , − , − , (0 , , , (0 , , − } + V and E = conv ( F ∪{ ( − , − , − , ( − , , − , (1 , , − , (1 , − , − } )+ V .In Figure 2.1 we can see upper faces of sets A, B, E, F ∈ C V ( R ), where V = { x ∈ R | x = x = 0 , x } , large dots represent the origin, andnumbers denote the third coordinate of vertices. It can be checked that A + F = B + E and that both pairs ( A, B ) and (
E, F ) are 0-minimal.0-10 -1 B . A . .. . -10-1 F -10-1-2-1 -1-2 E . Figure 2.1. Two equivalent minimal pairs of unbounded convex setsnot connected by translation from Example 2.9.Let us notice that if a given pair (
A, B ) does not have a property oftranslation and ( B − x ) ∩ ( − V ) = L V then a pair ( A − x, B − x ) may ormay not be 0-minimal. The following example shows such possibility. Example 2.10 (i). Let V = { (0 , } ⊂ R , A, B ∈ C V ( R ), B = conv { (0 , , (2 , } and A = conv ( B ∪ { (1 , } ). Let p ∈ B , p ∈ { x ∈ R | x min(1 − | x − | , } , p ∈ { x ∈ R | − | x − | < x < } .Denote B i = conv { (cid:0) ( B − p i ) ∪ { (0 , } (cid:1) , i = 0 , , A = A − p , A =conv { (cid:0) ( A − p ) ∪ { (0 , } (cid:1) , A = conv { (cid:0) ( A − p ) ∪ { (0 , , (1 , } (cid:1) if( p ) < A = conv { (cid:0) ( A − p ) ∪ { (0 , , ( − , } (cid:1) if ( p ) > A i , B i , i = 0 , , A . B B . A B . A Figure 2.2. Pairs of sets described in Example 2.10(i).All pairs of sets ( A i , B i ) , i = 0 , , A, B ). We aregoing to prove that each pair ( A i , B i ) is 0-minimal. Assume that( C, D ) ≺ ( A i , B i ) and 0 ∈ D ⊂ B i . By Theorem 7.4 a pair of poly-gons is minimal if and only if they have at most one pair of paralleledges that lie on the sam side of polygons. Then the pair of a triangleand a segment is minimal. Since the segment B contains 0, the pairs( A, B ) , ( A , B ) are minimal and 0-minimal. By Theorem 7.1, i.e. ex-istence of equivalent minimal pair contained in a given pair, and byTheorem 7.2, i.e. uniqueness-up-to-translation of equivalent minimalpairs of flat sets, the set D contains a translate of B , namely B − p i . JERZY GRZYBOWSKI AND RYSZARD URBA ´NSKI
Then obviously D = B i . Hence A i + B i = A i + D = B i + C , and bythe law of cancellation C = A i . Therefore, the pairs ( A i , B i ) , i = 0 , , A, B ]. Notice that ( B ) ∗ = B , ( B ) ∗ = ( B ) ∗ = { (0 , } . ByProposition 2.5 we obtain ( A ) ∗ = A , ( A ) ∗ = { (0 , } and ( A ) ∗ =conv { (0 , , (1 , } if ( p ) < A ) ∗ = conv { (0 , , ( − , } if( p ) > V ⊂ R be a cone such that { x ∈ V | x > } = { (0 , , } .Denote b A = ( A × { } ) + V and b B = ( B × { } ) + V . It can beproved that all 0-minimal pairs in [ b A, b B ] are ( b A i , b B i ) , i = 0 , , b A i = ( A i × { } ) + V and b B i = ( B i × { } ) + V and A i , B i are setsfrom (i). Notice that ( b B ) ∗ = B × { } and b B = ( b B ) ∗ + V , but( b B ) ∗ = { (0 , , } and b B = V = ( b B ) ∗ + V .
3. Reduced pairs of unbounded convex sets
Let us extend a notion of reduced pair of bounded sets from B ( R n )introduced by Bauer [5]. A pair ( A, B ) ∈ C V ( R n ) is reduced if [ A, B ] = { ( A + M, B + M ) | M ∈ C V ( R n ) } .In this section we show a relationship between reduced pairs and theproperty of translation of 0-minimal pairs. Proposition 3.1.
Let V be a closed convex cone. If a pair ( A, B ) ∈ C V ( R n ) is reduced then it has the property of translation.Proof. Let a pair (
C, D ) ∈ [ A, B ] be 0-minimal. Then (
C, D ) = ( A + M, B + M ) for some M ∈ C V ( R n ). Since 0 ∈ D = B + M , there exists b ∈ B such that − b ∈ M . Then A − b ⊂ A + M = C, B − b ⊂ B + M = D .Since ( C, D ) is 0-minimal, we obtain C = A − b, D = B − b . (cid:3) Proposition 3.2.
Let V ⊂ R n be a closed convex cone. If a pair ( A, B ) ∈ C V ( R n ) has the property of translation, then every -minimalpair ( C, D ) ∈ [ A, B ] is reduced .Proof. Let a pair (
C, D ) ∈ [ A, B ] be 0-minimal. Let b ∈ B . ByProposition 2.6(a) there exists 0-minimal pair ( C − x, D − x ) , x ∈ D ∗ such that C − x ⊂ A − b, D − x ⊂ B − b . We obtain D + b − x ⊂ B , and b − x ∈ B ˙ − D := { y | D + y ⊂ B } . Then b = x + ( b − x ) ∈ D + ( B ˙ − D ) ⊂ B , and B ⊂ D + ( B ˙ − D ) ⊂ B . Hence B = D + ( B ˙ − D ). Then A + D = D + ( B ˙ − D ) + C , and by the cancellation law A = C + ( B ˙ − D ). (cid:3) Propositions 3.1 and 3.2 can be summed up in the following theorem.
INIMAL PAIRS OF CONVEX SETS WHICH SHARE A RECESSION CONE 9
Theorem 3.3 (equivalence of reducibility and property of trans-lation).
Let V be a closed convex cone. A pair ( A, B ) ∈ C V ( R n ) isreduced if and only if this pair has the property of translation and is atranslate of some -minimal pair. Corollary 3.4.
Let V ⊂ R n be a closed convex cone. If a pair ( A, B ) ∈ C V ( R n ) has the property of translation, then ( B, A ) has the property oftranslation.Proof. If (
A, B ) has the property of translation then by Proposition3.2 some equivalent pair (
C, D ) is reduced, i.e. [
A, B ] = { ( C + M, D + M ) | M ∈ C V ( R n ) } . Hence the pair ( D, C ) is reduced and by Proposi-tion 3.1 the class [
B, A ] has the property of translation. (cid:3)
Theorem 3.3 shows a difference between the property of translation ofminimal pairs and the property of translation of 0-minimal pairs. Thereexists a broad class of minimal pairs of compact convex sets that satisfythe property of translation not being reduced pairs. For example allpairs of convex polygons (
A, B ) with exactly one pair ( A ( u ) , B ( u )) ofparallel edges (see Theorem 7.4). The authors can prove that if in a pair( A, B ) ∈ B ( R ) a set A is a tetrahedron and for all triangular faces A ( u ) the pairs ( A ( u ) , B ( u )) are minimal then ( A, B ) is minimal and hasthe property of translation. Such pair (
A, B ) may be a pair of convexpolyhedra and possess one or more pairs ( A ( v ) , B ( v )) of parallel edges.By Theorem 5.5, i.e. Bauer’s criterion for reduced polytopes the pair( A, B ) is not reduced. Sufficient and necessary condition for having theproperty of translation in R is not known. On the other hand every0-minimal pair having the property of translation of 0-minimal pairs isinevitably reduced.
4. Minimal pairs of unbounded planar convex sets
In order to prove propositions and theorems of this section we need topresent the notion and properties of an arc-length function f A corre-sponding to a planar set A . Let us consider a nonempty unboundedclosed convex set A ⊂ R . Let a recession cone V of A be pointedand unbounded . Obviously, V is a planar convex angle of a measure π − ϑ with ϑ ∈ (0 , π/ x -axisbisects the angle V . We construct an arc-length function f A followingthe approach from [18].Let u ∈ R and a support set of A in the direction of u be a set A ( u ) := { a ∈ A |h u, a i = max b ∈ A h u, b i} . Obviously, the support set A ( u ) is a singleton, a segment, a ray or an empty set. Let H A :( − ϑ, ϑ ) −→ bd A be a boundary function, where H A ( t ) is the center ofthe set A (cos t, sin t ), which is either a segment or a singleton. We alsodenote by f A : ( − ϑ, ϑ ) −→ R an arc length function of A , with a value f A ( t ) , t > A joining points H A (0) and H A ( t ). If t < f A ( t ) be op-posite to the length of the arc joining H A (0) and H A ( t ). The function f A is non-decreasing, f A (0) = 0 and f A ( t ) = ( f A ( t + ) + f A ( t − )) where f ( t + ) = lim s → t + f ( s ) , f ( t − ) = lim s → t − f ( s ), for t ∈ ( − ϑ, ϑ ).On the other hand, let f be any non-decreasing real function definedon an open interval ( − ϑ, ϑ ), such that f (0) = 0 and f ( t ) = 12 ( f ( t + ) + f ( t − )) , t ∈ ( − ϑ, ϑ ) . ( ∗∗ )We define the function H f : ( − ϑ, ϑ ) −→ R with the help of Stieltjesintegral H f ( t ) := t R ( − sin s, cos s ) df ( s ) , t > , − R t ( − sin s, cos s ) df ( s ) , t < . We denote A f := cl conv(im H f ) + V . Then we have A f A = A − H A (0)and f A f = f . The following proposition summarizes properties of thecorrespondence between non-decreasing functions and convex sets. Proposition 4.1.
Let
A, B ∈ C V ( R ) and f, g be non-decreasing func-tions satisfying ( ∗∗ ) . The following formulas hold true : f A + B = f A + f B , f tA = tf A for t > , A f + g = A f + A g , A tf = tA f for t > , f A g = g , A f B = B − H B (0) = B − midpoint B ( u ) , u = (1 , , f V ≡ , A f = V for f ≡ . Proposition 4.2 (criterion of planar summands).
A set A ∈ C V ( R ) is a summand of B ∈ C V ( R ) if and only if a function f B − f A is non-decreasing.Proof. If B = A + C then f B − f A = f C . On the other hand if a function g = f B − f A is non-decreasing then B − H B (0) = A − H A (0) + A g . (cid:3) The following proposition is needed in the proof of Theorem 5.2, i.e. acriterion for polytopal summands.
Proposition 4.3 (criterion of a polygonal summand).
Let P bea convex polygon, K ∈ C ( R ) . Assume that the recession cone of K isnot a straight line. Then P is a summand of K if and only if for all u ∈ S the support set K ( u ) is empty or contains a translate of P ( u ) .Proof. = ⇒ ) If K = P + L for some closed convex set L , and a face K ( u ) is nonempty then K ( u ) = P ( u ) + L ( u ), and K ( u ) contains atranslate of P ( u ). INIMAL PAIRS OF CONVEX SETS WHICH SHARE A RECESSION CONE 11 ⇐ =) If the cone V =recc K = K ˙ − K is a plane or a half-plane than thetheorem obviously holds true. Otherwise V is an angle of a measure π − ϑ , 0 < ϑ π/
2. We may assume that the x -axis is bisecting thecone V and that the negative part of the x -axis is contained in V .It is enough to show that P + V is a summand of K . Arc-length function f P + V is locally constant and noncontinuous only at t ∈ ( − ϑ, ϑ ) suchthat a support set ( P + V )( u ) , u = (cos t, sin t ) is a side of P . Sinceevery segment ( P + V )( u ), having length equal to f + P + V ( t ) − f − P + V ( t ),is contained in some translate of a segment K ( u ), of the lengh equalto f + K ( t ) − f − K ( t ), the difference of arc-length functions g = f K − f P + V is non-decreasing. By Proposition 4.2 the set P + V is a summand of K . (cid:3) Let us define the ordering of non-decreasing functions taking value 0at 0. For two functions f, g we say that f precedes g if and only if g − f is nondecreasing. Next two theorems on 0-minimal pairs in a planecorrespond to Theorem 3.1 and Corollary 3.2 from [18]. Theorem 4.4 (formula for an equivalent -minimal pair). Let ( A, B ) ∈ C V ( R ) . Denote g A := f A − inf( f A , f B ) and g B := f B − inf( f A , f B ) . Then the pair ( A g A + H A (0) − H B (0) , A g B ) is -minimaland belongs to [ A, B ] . Theorem 4.5 (criterion of 0-minimality).
Let ( A, B ) ∈ C V ( R ) .The pair ( A, B ) is minimal if and only if inf( f A , f B ) ≡ and ∈ B ( cost, sint ) for some t ∈ ( − θ, θ ) . Theorem 4.6 ( -minimal pair is reduced). Let V be a pointedunbounded convex cone in R . Then every -minimal pair ( A, B ) ∈ C V ( R ) is reduced.Proof. Let (
C, D ) ∈ [ A, B ]. Then A + D = B + C , f A + f D = f B + f C and H A (0) + H D (0) = H B (0) + H C (0). We have f C + inf( f A , f B ) ≺ inf( f C + f A , f C + f B ) and inf( f C + f A , f C + f B ) − f C ≺ inf( f A , f B ).Then f C + inf( f A , f B ) = inf( f C + f A , f C + f B ) = inf( f C + f A , f D + f A ) = f A + inf( f C , f D ). Hence g C := f C − inf( f C , f D ) = f A − inf( f A , f B ) = f A .In a similar way g D := f D − inf( f C , f D ) = f B − inf( f A , f B ) = f B . Thus( C, D ) = ( A f C + H C (0) , A f D + H D (0)) = ( A g C + A inf( f C ,f D ) + H C (0) , A g D + A inf( f C ,f D ) + H D (0)) = ( A f A + A inf( f C ,f D ) + H C (0) , A f B + A inf( f C ,f D ) + H D (0)) = ( A − H A (0) + A inf( f C ,f D ) + H C (0) , B − H B (0) + A inf( f C ,f D ) + H D (0)) = ( A + A inf( f C ,f D ) + H D (0) − H B (0) , B + A inf( f C ,f D ) + H D (0) − H B (0)). (cid:3)
5. Criterion for polytopal summands
In this section we generalize Shephard–Weil–Schneider criterion for apolytope being a summand of compact convex subset of R n . The fol-lowing Theorem 5.1 (Theorem 3.2.11. in [33]) was proved by Shephard[35] in the case of a polytope K and by Weil [37] in the case of compactconvex K . A strengthening of the theorem appeared in Grzybowski,Urba´nski and Wiernowolski [21]. Theorem. 5.1 (Shephard–Weil–Schneider criterion).
Let
P, K ∈ B ( R n ) , n > , P be a polytope. Then P is a summand of K if and onlyif the support set K ( u ) contains a translate of P ( u ) , whenever P ( u ) isan edge of P , u ∈ S n − . The next theorem, a generalization of Theorem 5.1 to an unboundedconvex set K , is based on Schneider’s proof from [32] presented inEncyclopedia of Mathematics and its Applications 151 [33]. Theorem 5.2 (criterion for a polytopal summand).
Let K ∈ C ( R n ) , n > , a recession cone V of K be pointed and P ⊂ R n be apolytope. Then P is a summand of K if and only if every nonemptybounded support set K ( u ) contains a translate of P ( u ) , whenever P ( u ) is an edge of P , u ∈ S n − .Proof. = ⇒ ) If a polytope P is a summand of K then there existsa set A ∈ C ( R n ) such that K = P + A . If a support set K ( u ) isnonempty then it is a Minkowski sum of respective support sets K ( u ) = P ( u ) + A ( u ). Hence K ( u ) contains a translate of P ( u ), whether P ( u )is an edge or not. ⇐ =) We are going to apply Minkowski duality between convex sets andsublinear functions. Basic facts on Minkowski duality are presented inSection 8. Since the cone V :=recc K is pointed, the effective domaindom h K has a nonempty interior. If a difference of support functions g := h K − h P is convex in the interior of dom h K then a function g = h K − h P is sublinear and lower semicontinuous. Then K = P + ∂g | ,and P is a summand of K . Hence we need to prove that the function g is convex over int dom h K . Notice that int V ◦ ⊂ dom h K ⊂ V ◦ , where V ◦ is a polar of the cone V .Let x, y ∈ int dom h K . If 0 lies between x and y then 0 ∈ int dom h K and dom h K = R n . Hence V = { } . This is true only if K is bounded.In this case the polytope P is a summand of K by Theorem 5.1.Otherwise, lin { x, y } is a two-dimensional subspace of R n . Let pr: R n −→ lin { x, y } be a perpendicular projection. Images pr K and pr P of K and P by projection pr are two-dimensional convex sets. For any z ∈ lin { x, y } equalities h K ( z ) = h pr K ( z ) and h P ( z ) = h pr P ( z ) hold truefor respective support functions. Assume that every side of the con-vex polygon pr P , that is (pr P )( u ), u ∈ lin { x, y } is equal to an image INIMAL PAIRS OF CONVEX SETS WHICH SHARE A RECESSION CONE 13 pr( P ( u )) of a single edge P ( u ) of the polytope P . It simply meansthat the support set P ( u ) is an edge of P . Then if a set (pr K )( u ) =pr( K ( u )), u ∈ lin { x, y } is nonempty then (pr K )( u ) contains a trans-late of pr( P ( u )) = (pr P )( u ) since K ( u ) contains a translate of P ( u ).Hence by Proposition 4.3 the set pr P is a summand of pr K . Thus h pr K − h pr P is a convex function, and the function g = h K − h P re-stricted to lin { x, y } is also convex. Therefore, g ( x + y ) g ( x )+ g ( y )2 . If not every side of pr P is equal to a projection of a single edge of P then still there exists a sequence ( y n ) tending to y such that anyside of polygon pr n P , where pr n is a perpendicular projection onto thesubspace lin { x, y n } , is equal to a projection of single edge P ( u ) of P .Since the function g = h K − h P is continuous in the interior of dom h K ,we obtain g (cid:18) x + y (cid:19) = lim n −→∞ g (cid:18) x + y n (cid:19) lim n −→∞ g ( x ) + g ( y n )2 = g ( x ) + g ( y )2 . Since g is continuous in int dom h K and x, y are arbitrary, we have justproved that g is convex in int dom h K . On the other hand g = h K − h P is lower semicontinuous, hence convex in all R n . Therefore, by Theorem8.1, we obtain K = P + ∂g | . (cid:3) Remark 5.3.
Notice that in Theorem 5.2 the assumption of recessioncone being pointed is necessary. For example let K be a straight line in R n and let P be any polytope not contained in a straight line parallelto K . Then P is not a summand of K . However, if a support set K ( u )is not empty then K ( u ) = K , and K ( u ) is unbounded.Let us extend a notion of polytope to unbounded sets sharing a pointedrecession cone V . By P V ( R n ) := { P + V | P ∈ P ( R n ) } , where P ( R n ) isa family of all nonempty polytopes in R n , we denote the family of sumsof polytopes and the cone V . We call elements of the family P V ( R n )by V -polytopes . V -polytope is the smallest convex set with a recessioncone V containing a given finite set of points. The following theoremis straightforward corollary from Theorem 5.2. Theorem 5.4 (criterion for a V -polytopal summand). Let V bea pointed convex cone, K ∈ C V ( R n ) , n > , and P ∈ P V ( R n ) . Then P is a summand of K if and only if a nonempty bounded support set K ( u ) contains a translate of P ( u ) , whenever P ( u ) is an edge of P , u ∈ S n − . Let
A, B ∈ C ( R n ). We call two bounded support sets A ( u ) and B ( u ) equiparallel edges if they are parallel line segments. Bauer in [5] gavethe following necessary and sufficient criterion for reduced pairs of poly-topes. Theorem 5.5 (Bauer’s criterion for reduced pair of polytopes).
A pair ( A, B ) of polytopes in R n is reduced if and only if A and B haveno equiparallel edges. The next theorem generalizes Bauer’s criterion to reduced pairs of V -polytopes. Theorem 5.6 (criterion for reduced pair of V -polytopes). Let V be a pointed convex cone. Then a pair ( A, B ) ∈ P V ( R n ) is reducedif and only if A and B have no equiparallel edges.Proof. ⇐ =) Let A and B have no equiparallel edges. Assume that A + D = B + C =: E for some C, D ∈ C V ( R n ). In order to prove that A + B is a summand of E , let ( A + B )( u ) be an edge. Since A and B have no equiparallel edges, A ( u ) and B ( u ) cannot be line segmentsboth at the same time. Then one of these, say B ( u ), is a singleton and( A + B )( u ) is a translate of A ( u ). Hence the set E ( u ) = A ( u ) + D ( u )contains a translate of ( A + B )( u ). By Theorem 5.4, the set A + B isa summand of E = A + D = B + C . Therefore, E = A + B + M forsome M ∈ C V ( R n ). By the cancellation law ( C, D ) = ( A + M, B + M ).= ⇒ ) If A ( u ) and B ( u ) are parallel edges then we can construct a pair( A ′ , B ′ ) equivalent to ( A, B ) such that A ⊂ A ′ , B ⊂ B ′ and no translateof A ( u ) is contained in A ′ ( u ). This construction was given by Bauer inTheorem 5.3 [5] for a pair of polytopes. (cid:3)
6. Application. Minimal representation of a difference ofconvex functions
Let V ⊂ R n +1 be a nontrivial closed convex cone such that V ∩ { x ∈ R n +1 | x n +1 > } = { } . A pair ( A, B ) ∈ C V ( R n +1 ) is H - minimal if ( A, B ) is a minimal element in the family { ( C, D ) ∈ [ A, B ] | ∈ D and ∀ x ∈ D : x n +1 } . The definition of H -minimality corre-sponds to Hartman’s [22] definition of a minimal representation of adc-function f = g − h , i.e. a difference of convex functions g and h ,defined on the open unit ball in R n .Let us notice that for two convex and lower semicontinous functions g, h : R n −→ R ∪ { + ∞} we can find corresponding cosed convex sets A, B such that g ( x ) = h A ( x, , h ( x ) = h B ( x, , x ∈ R n . The sets
A, B are defined by A := { ( x, t ) ∈ R n × R | ∀ y : h ( x, t ) , ( y, i g ( y ) } = { ( x, t ) |∀ y : h x, y i + t g ( y ) } , ( ∗ ∗ ∗ ) B := { ( x, t ) |∀ y : h x, y i + t h ( y ) } . INIMAL PAIRS OF CONVEX SETS WHICH SHARE A RECESSION CONE 15
Indeed, by Theorem 8.2 the set A is a subdifferential of such a lowersemicontinuous sublinear function ˆ g , that ˆ g ( x, t ) = tg ( x/t ) , t > g ( x, t ) = + ∞ , t <
0. In fact A = hypo ( − g ∗ ) i.e. the convex set A isequal to a hypograph of a function − g ∗ where g ∗ is a convex conjugateof g [31]. We also have B = hypo ( − h ∗ ).Hartman, defining minimal representation of a dc-function f = g − h in section 6 of [22], requires that g, h are as small as possible underconditions of h h (0) = 0. The function h is non-negative ifand only if 0 R n +1 ∈ B . Besides, h (0) B ⊂ R n × R − . If B ⊂ R n × R − then h (0) = h B (0 ,
1) = sup ( x,t ) ∈ B h ( x, t ) , (0 , i = sup ( x,t ) ∈ B t f = g − h defined on an interior of aunit ball B in R n . In order to represent convex functions g, h by convexsets we extend them outside of int B by g ( x ) = h ( x ) = ∞ for x B and g ( x ) = lim inf y → x, k y k < g ( y ), h ( x ) = lim inf y → x, k y k < h ( y ) for k x k = 1.Since effective domains of g and h contain an open Euclidean unit balland are contained in a closed unit ball B , the sets A and B sharerecession cone V defined by V = { ( x, t ) ∈ R n × R | t −k x k } . Fromprevious considerations follows the next theorem. Theorem 6.1.
A representation f = g − h of a dc -function is minimalaccording to Hartman if and only if a pair of sets ( A, B ) , where A, B are defined by ( ∗ ∗ ∗ ) , is H -minimal in C V ( R n +1 ) . If we replace in Hartman’s definition an open unit ball with an interiorof a closed convex set K containing 0 then corresponding sets A and B share a recession cone V defined by V := S t > t ( K ◦ × {− } ) where K ◦ is a polar of K .The following proposition is obvious. Proposition 6.2.
A pair ( A, B ) ∈ C V ( R n +1 ) is H -minimal if and onlyif it is -minimal and B ⊂ { x ∈ R n +1 | x n +1 } . In Example 2.10(ii) all equivalent 0-minimal pairs are H -minimal. Ob-viously, 0-minimal pairs may not be H -minimal. See the next example. Example 6.3.
Let T : R −→ R , T ( x , x , x ) := ( x , x , sx + tx + x ) , s, t ∈ R . Consider convex sets from Example 2.10(ii). For i =0 , , T ( b A i ) , T ( b B i )) are 0-minimal. The pair ( T ( b A i ) , T ( b B i ))is H -minimal if and only if − s ( p i ) − t ( p i ) s (2 − ( p i ) ) − t ( p i )
0. For example the pair ( T ( b A ) , T ( b B )) where T ( x , x , x ) :=( x , x , x + x ), p = (0 ,
0) is 0-minimal and not H -minimal. Obviously, any pair which is 0-minimal and not H -minimal does notcontain a H -minimal pair. Theorem 6.4 (existence of H -minimal pairs). Let ( A, B ) ∈ C V ( R n +1 ) . There exists an equivalent H -minimal pair ( A ′ , B ′ ) suchthat A ′ ⊂ A − b, B ′ ⊂ B − b for some b ∈ B .Proof. Let b ∈ B and b n +1 = max x ∈ B x n +1 . By Theorem 2.1 there exists a0-minimal pair ( A ′ , B ′ ) contained in ( A − b, B − b ). Since B ′ ⊂ B − b ⊂{ x ∈ R n +1 | x n +1 } , the pair ( A ′ , B ′ ) is H -minimal. (cid:3) Remark 6.5.
It is possible that among equivalent pairs of sets a H -minimal pair is unique even if this pair does not have the property oftranslation. For example the pair ( T ( b B ) − (1 , , , T ( b A ) − (1 , , T ( x , x , x ) := ( x , x , x + x ), p = (0 , H -minimal pair in the quotient class [ T ( b B ) , T ( b A )]. Noticethat T ( b A ) − (1 , ,
1) = conv { (0 , , , ( − , − , − , (1 , − , − } + V , T ( b B ) − (1 , ,
1) = conv { ( − , − , − , (1 , − , − } + V . Convex func-tions corresponding to these two sets are g ( x , x ) := | x | − x − h ( x , x ) := max(0 , | x | − x − f ( x , x ) := min(0 , | x | − x −
1) = g ( x , x ) − h ( x , x ). Proposition 6.6.
Let a pair ( A, B ) ∈ C V ( R n +1 ) be reduced and V ∩{ x ∈ R n +1 | x n +1 > } = L V . Then a pair ( A − x, B − x ) , x ∈ B is H -minimal if and only if x n +1 = sup y ∈ B y n +1 = h B ( u ) , where u =(0 , ..., , ∈ R n +1 .Proof. By Theorem 3.3 the pair (
A, B ) has property of translation.Proposition follows from criterion of 0-minimality in Proposition 2.6(b)and from characterization of H -minimality in Proposition 6.2. (cid:3) The following example shows, that reducibility (property of transla-tion) in the assumptions of Proposition 2.6(b) is essential.
Example 6.7.
Consider a pair ( b A , b B ) in Example 2.10(ii). This pairis H -minimal but not reduced. If x ∈ b B then ( b A − x, b B − x ) is H -minimal if and only if x = 0. Still ( b A − x, b B − x ) is H -minimal if andonly if x ∈ ( b B ) ∗ , i.e. x = ( x , ..., x n +1 ) ∈ b B and x n +1 = sup y ∈ b B y n +1 = 0.
7. Appendix. Minimal pairs of closed bounded convex sets
Let B ( R n ) be a family of all nonempty compact convex sets, i.e. convexbodies. The idea of treating compact convex sets as numbers or, rather,as vectors goes back to Minkowski [24]. A semigroup of nonemptybounded closed convex subsets B ( X ) of a vector space X was embedded INIMAL PAIRS OF CONVEX SETS WHICH SHARE A RECESSION CONE 17 into a topological vector space in the case of a normed space X byR˚adstr¨om [29], a locally convex space X by H¨ormander [23] and atopological vector space by Urba´nski [36]. The embedding was possiblethanks to an order cancellation law: A + B ⊂ cl ( B + C ) = ⇒ A ⊂ C for A, B, C ∈ B ( X ) . For a concise proof of an order cancellation law in a more general set-ting we refer the reader to Proposition 5.1 in [14]. Convex sets are em-bedded into Minkowski–R˚adstr¨om–H¨ormander space e X = B ( X ) / ∼ ofquotient classes, where a relation of equivalence is defined by ( A, B ) ∼ ( C, D ) : ⇐⇒ cl ( A + D ) = cl ( B + C ).A new motivation to study pairs of convex sets came from quasidifferen-tial calculus of Demyanov and Rubinov [8, 9], where a quasidifferential Df ( x ) is a pair of convex sets ( A, B ) = ( ∂f | x , ∂f | x ) called sub- andsuperdifferential. Rather than a pair of sets ( A, B ) a quasidifferentialis a quotient class [
A, B ] := [(
A, B )] ∼ .The best representation of a quotient class [ A, B ] is a reduced pair , i.e.a pair (
A, B ) such that [
A, B ] = { ( A + C, B + C ) | C ∈ B ( X ) } . Then alltranslates of ( A, B ) give all minimal elements of [
A, B ]. Reduced pairswere studied by Bauer [5]. However, not every quotient class [
A, B ]contains a reduced pair. We say that a pair (
A, B ), or a quotientclass [
A, B ] has property of translation if all minimal pairs in [
A, B ] aretranslates of each other. There exist not reduced minimal pairs thathave property of translation. The following theorem holds true.
Theorem 7.1. ([19, 26])
Let X be a reflexive Banach space. For everypair ( A, B ) ∈ B ( X ) , there exists an inclusion-minimal pair ( C, D ) ∈ [ A, B ] such that C ⊂ A, D ⊂ B . Caprari and Penot [7] proved existence of inclusion minimal pairs in aquotient class [
A, B ] ∈ C ( X ) × K ( X ) / ∼ where K ( X ) is a family of allnonempty compact convex subsets of a locally convex vector space X . Theorem 7.2. ([5, 12, 34])
Let ( A, B ) ∈ B ( R ) . A minimal pair in [ A, B ] is unique up to translation. Theorem 7.2 basically states that every minimal pair of two-dimensionalcompact convex sets has property of translation.
Example 7.3. ([12]) In R we have equivalent minimal pairs notconnected by translation . A B C ❜❜ ✧✧
D E F
Figure 7.1. Three equivalent minimal pairs not connected bytranslation.In Figure 7.1 the solid B is a regular octahedron, D is an elongatedoctahedron, F is a hexagon, A is a rhombohedron and E is a cubocta-hedron.In [13, 15, 27] more quotient classes [ A, B ] with no unique minimal pairwere found in R . However, the set of all equivalent minimal pairs wasnever effectively described for a quotient class [ A, B ] with no uniqueminimal element. All these results enabled calculus of pairs of convexsets in a way analogous to fractional arithmetics [28].The following theorem states a necessary and sufficient criterion forminimal pairs of convex polygons.
Theorem 7.4 (Theorem 3.5 in [16] ). A pair ( A, B ) of flat polytopesis minimal if and only if A and B have at most one pair of paralleledges that lie on the same side of the polytopes.
8. Appendix. Minkowski duality
In this section we present Minkowski duality between closed convexsets and sublinear functions.A 4-tuple ( X, R + , + , · ), where an operation of addition ’+’ and of mul-tiplication by nonnegative numbers ’ · ’ are defined for elements of theset X , is called an abstract convex cone if (1) the pair ( X, +) is acommutative group and for all x, y ∈ X and all s, t > x = x , (3) 0 x = 0, (4) s ( tx ) = ( st ) x , (5) t ( x + y ) = tx + ty and (6)( s + t ) x = sx + tx .If a set A belongs to the family C ( R n ) of all nonempty closed convexsubsets of R n then its support function h A is defined by h A := sup a ∈ A h a, ·i . Minkowski addition A ˙+ B of sets belonging to C ( R n ) is defined by A ˙+ B := cl( A + B ). Obviously, if one of these sets is bounded then A ˙+ B = A + B .If a function h : R n −→ R ∪ {∞} belongs to the family S ∞ lsc ( R n ) of allsublinear (positively homogenous and convex) lower semicontinuous INIMAL PAIRS OF CONVEX SETS WHICH SHARE A RECESSION CONE 19 functions then its subdifferential at 0 is a closed convex set defined by ∂h | := { x ∈ R n | h x, ·i h } .Both 4-tuples ( C ( R n ) , R + , ˙+ , · ) and ( S ∞ lsc ( R n ) , R + , + , · ) are abstract con-vex cones and Minkowski duality establishes isomorphic relationshipbetween these cones. Theorem 8.1.
The mapping S ∞ lsc ( R n ) ∋ h ∂h | ∈ C ( R n ) is anisomorphic bijection from an abstract convex cone S ∞ lsc ( R n ) onto an ab-stract convex cone C ( R n ) . The mapping C ( R n ) ∋ A h A ∈ S ∞ lsc ( R n ) is an inverse mapping. Moreover, a restriction of the mapping to thesubfamily of finite sublinear functions S ( R n ) is an isomorphic bijectionfrom a subcone S ( R n ) onto a subcone B ( R n ) of all nonempty boundedclosed convex sets. Theorem 8.1 is stated in [17] in a general case for a dual pair (
X, Y )of linear spaces over R where h· , ·i is such a bilinear function, thatfunctions {h y, ·i} y ∈ Y separate points in X and functions {h· , x i} x ∈ X separate points in Y .Let h : R n −→ R ∪ {∞} be a sublinear lower semicontinous function.An effective domain dom h ⊂ R n is a convex cone, its closure cl dom h is a closed convex cone.Let V be a closed convex cone in R n . A characteristic function χ V is de-fined by χ V ( x ) := (cid:26) , x ∈ V, ∞ , x V. A subdifferential of χ V at 0 coincideswith a polar cone V ◦ . Therefore, ∂ ( χ V ) | = V ◦ , and h V = χ V ◦ .By S ∞ lsc,V ( R n ) we denote a subfamily of sublinear functions with finitevalues in the relative interior of V and infinite outside of V . Valuesof such a function on the relative boundary of V are determined byits values in the relative interior. A 4-tuple ( S ∞ lsc,V ( R n ) , R + , + , · ) isan abstract convex cone after modifying multiplication by 0 in thefollowing way 0 h =: χ V .By C V ( R n ) we denote all closed convex sets A having their recessioncone recc A := A ˙ − A = { x | x + A ⊂ A } equal to V . Again the family C V ( R n ) is an abstract convex cone after modifying multiplication by 0with a formula 0 A = V [30]. Theorem 8.2.
The mapping S ∞ lsc,V ( R n ) ∋ h ∂h | ∈ C V ◦ ( R n ) is anisomorphic bijection from an abstract convex cone S ∞ lsc,V ( R n ) onto anabstract convex cone C V ◦ ( R n ) . The mapping C V ◦ ( R n ) ∋ A h A ∈ S ∞ lsc,V ( R n ) an is an inverse mapping. Proof.
In view of Theorem 8.1 it is enough to prove that for any func-tion h ∈ S ∞ lsc ( R n ) closed convex cones V =cl dom h and V =recc( ∂h | )are mutually polar.Notice that h = h + χ V . By Theorem 8.1 we obtain ∂h | = ∂h | + ∂ ( χ V ) | = ∂h | + V ◦ . Then V ◦ ⊂ ∂h | ˙ − ∂h | =recc( ∂h | ) = V .On the other hand ∂h | = ∂h | + V . By Theorem 8.1 we get h = h + h V = h + χ V ◦ . Hence dom h ⊂ V ◦ , and V ⊂ V ◦ . Therefore, V ⊂ V ◦ . (cid:3) References [1] M.E. Abbasov,
Comparison Between Quasidifferentials and Exhausters , J. Op-tim. Theory Appl., 175 (1) (2017), 59–75.[2] T. Antczak,
Optimality conditions in quasidifferentiable vector optimization ,J. Optim. Theory Appl., 171 (2016), 708–725.[3] M.V. Balashov, E.S. Polovinkin,
M-strongly convex subsets and their generat-ing sets , Sbornik. Math. 191 (2000), 25–60.[4] E.K. Basaeva, A.G. Kusraev, S.S. Kutateladze,
Quasidifferentials in Kan-torovich spaces , J. Optim. Theory Appl., 171 (2016), 365–383.[5] Ch. Bauer,
Minimal and reduced pairs of convex bodies , Geom. Dedicata 62(1996), 179–192.[6] J. Bielawski, J. Tabor,
An embedding theorem for unbounded convex sets in aBanach space , Demonstr. Math. 42 (4) (2009), 703–709.[7] E. Caprari, J-P. Penot,
Tangentially ds functions , Optimization 56 (1-2)(2007), 25–38.[8] V.F. Demyanov and A.M. Rubinov,
Quasidifferential Calculus , OptimizationSoftware Inc., Springer–Verlag, New York, 1986.[9] V.F. Demyanov and A.M. Rubinov,
Quasidifferentiability and Related Topics ,Nonconvex Optimization and its Applications, Kluwer Academic. Publisher,Dortrecht–Boston–London, 2000.[10] M.V. Dolgopolik
A New Constraint Qualification and Sharp Optimality Con-ditions for Nonsmooth Mathematical Programming Problems in Terms of Qua-sidifferentials , SIAM J. Optim., 30 (3) (2020), 2603–2627.[11] L. Drewnowski,
Additive and countably additive correspondences , Comment.Math. 19 (1976), 25–54.[12] J. Grzybowski,
Minimal pairs of compact sets , Arch. Math. 63 (1994), 173–181.[13] J. Grzybowski, S. Kaczmarek and R. Urba´nski,
General methods of construct-ing equivalent minimal pairs not unique up to translation , Rev. Mat. Complut.13 (2) (2000), 383–398.[14] J. Grzybowski, M. K¨u¸c¨uk, Y. K¨u¸c¨uk, R. Urba´nski.
Minkowski–R˚adstr¨om–H¨ormander cone , Pac. J. Optim. 10 (4) (2014), 649–666.[15] J. Grzybowski, D. Pallaschke and R. Urba´nski,
Minimal pairs representingselections of four linear functions in R , J. Convex Anal. 7 (2) (2000), 445–452.[16] J. Grzybowski, D. Pallaschke and R. Urba´nski, Minimal pairs ofbounded closed convex sets as minimal representations of elements of theMinkowski–R˚adstr¨om–H¨ormander spaces , Banach Center Publications 84(2009), 31–55.[17] J. Grzybowski, D. Pallaschke and R. Urba´nski,
Support functions and subdif-ferentials , Comment. Math. 56 (1) (2016), 45–53.
INIMAL PAIRS OF CONVEX SETS WHICH SHARE A RECESSION CONE 21 [18] J. Grzybowski, H. Przybycie´n,
Minimal Representation in a Quotient Spaceover a Lattice of Unbounded Closed Convex Sets , J. Convex Anal. 24 (2) (2017),695–705.[19] J. Grzybowski, R. Urba´nski,
Minimal pairs of bounded closed convex sets , Stu-dia Math. 126 (1997), 95–99.[20] J. Grzybowski, R. Urba´nski,
Order cancellation law in the family of boundedconvex sets , J. Global Optim. 77 (2020), 289–300.[21] J. Grzybowski, R. Urba´nski, M. Wiernowolski,
On common summands andantisummands of compact convex sets , Bull. Polish. Acad. Sci. Math. 47 (1999),69–76.[22] Ph. Hartman,
On functions representable as a difference of convex functions ,Pac. J. Math. 9 (1959), 707–713.[23] L. H¨ormander,
Sur la fonction d’appui des ensembles convexes dans un espacelocalement convexe , Arkiv Math. 3 (1954), 181–186.[24] H. Minkowski,
Theorie der konvexen K¨orper, insbesondere Begr¨undung ihresOberfl¨achenbegriffs.
Gesammelte Abhandlungen, vol. II, 131–229, Teubner,Leipzig, 1911.[25] G.M. Moln´ar, Z. P´ales,
An extension of the R ˚a dstr¨om cancellation theoremto cornets , arXiv:2008.01408, 2020.[26] D. Pallaschke, S. Scholtes, R. Urba´nski,
On minimal pairs of convex compactsets , Bull. Polish Acad. Sci. Math. 39 (1991), 1–5.[27] D. Pallaschke, R. Urba´nski,
A continuum of minimal pairs of compact convexsets which are not connected by translations , J. Convex Anal. 3 (1996), 83–95.[28] D. Pallaschke, R. Urba´nski,
Pairs of Compact Convex Sets, Fractional Arith-metic with Convex Sets , Mathematics and Its Applications, Kluwer AcademicPublisher, Dortrecht–Boston–London, 2002.[29] H. R˚adstr¨om,
An embedding theorem for spaces of convex sets , Proc. Amer.Math. Soc. 3 (1952), 165–169.[30] S. Robinson,
An embedding theorem for unbounded convex sets , Technical Sum-mary Report No. 1321, Madison Mathematics Research Center, University ofWisconsin (1973) 1–23.[31] R.T. Rockafellar,
Convex Analysis , Princeton Univ. Press, Princeton, N.J.1972.[32] R. Schneider,
Summanden konvexer K¨orper , Arch. Math. 25 (1974), 83–85.[33] R. Schneider,
Convex Bodies: The Brunn-Minkowski Theory , Encyclopediaof Mathematics and its Applications, 151. Cambridge University Press, Cam-bridge 2014.[34] S. Scholtes,
Minimal pairs of convex bodies in two dimensions , Mathematika39 (1992), 267–273.[35] G.C. Shephard,
Decomposable convex polyhedra , Mathematika 10 (1963), 89–95.[36] R. Urba´nski,
A generalization of the Minkowski–R˚adstr¨om–H¨ormander Theo-rem , Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 24 (1976), 709–715.[37] W. Weil,
Decomposition of convex bodies , Mathematika 21 (1974), 19–25.[38] L.-W. Zhang, Z.-Q Xia, Y. Gao and M.-Z. Wang,
Star-kernels and star-differentials in quasidifferential analysis , J. Convex Anal. 9 (1) (2002), 139–158.
Faculty of Mathematics and Computer Science, Adam MickiewiczUniversity, Uniwersytetu Pozna´nskiego 4, 87, 61-614 Pozna´n, Poland
Email address : [email protected] Faculty of Mathematics and Computer Science, Adam MickiewiczUniversity, Uniwersytetu Pozna´nskiego 4, 87, 61-614 Pozna´n, Poland
Email address ::