Author mnuscript, published in "SIAM Journl on Optimiztion 18, 3 2007) 834-852" DOI : 10.1137/060658047 SIAM J. OPTIM. Vol. 0, No. 0, pp. 000 000 c XXXX Society for Industril nd Applied Mthemtics BOUNDARY HALF-STRIPS AND THE STRONG CHIP EMIL ERNST AND MICHEL THÉRA Abstrct. When the subdifferentil sum rule formul holds for the indictor functions ι C nd ι D of two closed convex sets C nd D of loclly convex spce X, the pir C, D) is sid to hve the strong conicl hull intersection property the strong CHIP). The specifiction of well-known theorem due to Moreu to the cse of the support functionls σ C nd σ D subsumes the fct tht the pir C, D) hs the strong CHIP whenever the inf-convolution of σ C nd σ D is exct. In this rticle we prove, in the setting of Eucliden spces, tht if the pir C, D) hs the strong CHIP while the boundry of C does not contin ny hlf-strip, then the inf-convolution of σ C nd σ D is exct. Moreover, when the boundry of closed nd convex set C does contin hlf-strip, it is possible to find closed nd convex set D such tht the pir C, D) hs the strong CHIP while the inf-convolution of σ C nd σ D is not exct. The vlidity of the converse of Moreu s theorem in Eucliden spces is thus ssocited to the bsence of hlf-strips within the boundry of concerned convex sets. Key words. strong conicl hull intersection property, convex progrmming with convex inequlities, Eucliden spce, exct infiml convolution, qulifiction conditions AMS subject clssifictions. 90C46, 90C51, 46N10, 49K40 DOI. 10.1137/060658047 1. Introduction. This study concerns n ppliction of geometricl notion clled the strong conicl hull intersection property strong CHIP) introduced by Deutsch, Li, nd Swetits see [6]). We sy tht the pir C, D) of closed nd convex subsets of some loclly convex spce X hs the strong CHIP if the subdifferentil of the sum nd sum of the subdifferentils of their indictor functions coincide: 1.1) ι C + ι D )= ι C + ι D. As customry, ι A is the indictor function of subset A of X nd is defined by ι A x) =0ifx A, nd ι A x) =+ otherwise. We lso recll tht the convex subdifferentil is n opertor from X into the topologicl dul X of X, which ssigns to ech extended-rel-vlued mpping Φ on X set-vlued opertor between X nd X defined by Φx 0 )={x X : x x 0,x +Φx 0 ) Φx) x X}, where, : X X R is the dulity piring between X nd X. Reclling tht the norml cone N C x) tx X to closed convex set C of X is the set ι C x), the strong CHIP for the pir C, D) mounts to sying tht 1.2) N C D x) =N C x)+n D x) x C D; i.e., every norml direction to C D t some point x cn be expressed s the sum of norml directions t x to C nd D. Received by the editors April 24, 2006; ccepted for publiction in revised form) Jnury 16, 2007; published electroniclly DATE. http://www.sim.org/journls/siopt/x-x/65804.html Aix-Mrseille Université, EA2596, Mrseille, F-13397, Frnce Emil.Ernst@univ.u-3mrs.fr). XLIM, Université de Limoges, 123 Avenue A. Thoms, 87060 Limoges Cedex, Frnce michel. ther@unilim.fr). 1
2 EMIL ERNST AND MICHEL THÉRA This property is importnt in convex optimiztion becuse when we consider the problem of minimizing convex functionl Φ on the intersection of two sets C nd D which hve the strong CHIP, the optimlity condition for x to be minimizer becomes 0 Φ x)+n C x)+n D x). In the cse of convex differentible optimiztion, it becomes f x) N C x)+n D x). Let us quote only the result proved when X is Hilbert spce by Deutsch see [4]). It sys tht the strong CHIP is the wekest constrint qulifiction under which minimizer x of convex function Φ : C 1 C 2 R cn be chrcterized using the subdifferentil of Φ t x nd the norml cones of C 1 nd C 2 t x. The existence of conditions ensuring tht pir of closed nd convex sets hs the strong CHIP is bsed on clssicl result by Moreu [9, Remrque 10.2]); the result initilly published in [8] mkes use of nother key concept of convex nlysis, nmely the notion of infiml convolution inf-convolution). Recll tht, if Φ nd Ψ re extended-rel-vlued lower semicontinuous convex functions over X this clss of functions from now on is denoted by Γ 0 X)), the inf-convolution of Φ nd Ψ is the extended rel-vlued function Φ Ψ defined by 1.3) Φ Ψx) = inf Φx y)+ψy)). y X The infiml convolution between Φ nd Ψ is sid to be exct if Φ Ψ Γ 0 X) nd the infimum is chieved in 1.3) whenever Φ Ψx) < +. Let us lso recll tht given convex closed set A in X, we note σ A : X R {+ }, the support function of A. It is defined by σ A f) = sup x, f. x A Using this concept, Moreu s theorem sttes tht the subdifferentil sum formul 1.1) holds, provided tht the inf-convolution of the support functionls of C nd D is exct. Remrk tht n equivlent wy of expressing the exctness of the inf-convolution of the support functionls σ C nd σ D is to sy tht every liner functionl f X which is bounded bove on C D my be expressed s the sum of two liner functionls f 1 nd f 2, bounded bove on C nd D respectively, such tht sup x C D x, f = sup x C x, f 1 + sup x, f 2. x D Let us observe tht reltion 1.2), i.e., the strong CHIP, is equivlent to the following property: Every liner functionl f X which chieves its mximum on C D my be expressed s the sum of two liner functionls f 1 nd f 2, chieving their mximum on C nd D, respectively, such tht mx x, f = mx x, f 1 + mx x, f 2. x C D) x C x D The importnce of Moreu s theorem comes from the fct tht severl very generl qulifiction conditions re known to ensure the exctness of the inf-convolution of support functionls the reder is referred for further informtion to the excellent
BOUNDARY HALF-STRIPS AND THE STRONG CHIP 3 rticles of Zălinescu [13], Gowd nd Teboulle [10], nd, respectively, Simons [11], in which he/she my find cler picture of the topic, s well s self-contined proofs for most of the concerned results). Accordingly, the result proved by Moreu gives the possibility of systemticlly specifying every qulifiction condition s criterion for the strong CHIP see for instnce [5, Proposition 2.3].) Let us remrk tht the exctness of the inf-convolution of the support functionls is stronger thn the simple strong CHIP. Indeed, Moreu s condition concerns ll the continuous liner functionls bounded bove on the intersection C D, while the strong CHIP is formulted only in terms of those elements from X which chieve their mximum on C D. The question is thus rised of the vlidity of the converse to this theorem. The converse to Moreu s theorem obviously holds for sets C nd D such tht every liner nd continuous mp bounded bove on ny one of the sets C D, C, or D necessrily chieve their mximums on this set. On this ground, first prtil converse of the Moreu result hs recently been proved by Buschke, Borwein, nd Li for Hilbert spces see [1, Proposition 6.4]); the result ws extended to the setting of Bnch spces by Burchik nd Jeykumr [3, Proposition 4.2]. Their result sttes tht, if C nd D is pir of closed nd convex cones with the strong CHIP, then the inf-convolution of their support functionls is lwys exct. However, it is not necessry to impose to every liner nd continuous mp which is bounded bove on ny one of the sets C D, C, ord, to chieve its mximum on this set, in order to ensure the vlidity of the converse to the bove mentionned Moreu s theorem. It is the im of this rticle to clerly define the best conditions under which the converse of Moreu s theorem holds. More precisely, we chrcterize ll the closed nd convex subsets C of n Eucliden spce X such tht the following converse of Moreu s theorem holds: If, for some closed nd convex set D, the pir C, D) hs the strong CHIP, then the inf-convolution of σ C nd σ D is exct. Our min result sttes tht the vlidity of the converse of Moreu s theorem is ensured if nd only if the boundry of the closed nd convex subset C of the Eucliden spce X does not contin ny hlf-strip by hlf-strip we men, s customry, the convex hull of two disjoint nd prllel hlf-lines). Note tht the clss of closed nd convex sets without boundry hlf-strips is rther lrge, s it contins the list is not exhustive ll the bounded sets, the strictly convex sets, or even the continuous sets in the sense of Gle nd Klee sets such tht their support functionl is continuous except t the origin see [7]). The outline of the pper is s follows. The cse of closed nd convex sets without boundry hlf-strips is considered in section 2. We prove Theorem 2.3) tht, if the pir C, D) hs the strong CHIP, nd if the boundry of one of the sets, sy C, does not contin ny hlf-strip, then the inf-convolution of the support functions of C nd D must be exct. The lst section is concerned with convex sets which do dmit t lest one boundry hlf-strip. If the boundry of closed nd convex set C contins some hlf-strip, then we give construction of closed nd convex set D such tht the pir C, D) hs the strong CHIP, while the inf-convolution of σ C nd σ D fils to be exct. 2. Convex sets without boundry hlf-strips. Now let us first collect some conditions ensuring in every reflexive Bnch spce the vlidity of the converse of Moreu s theorem.
4 EMIL ERNST AND MICHEL THÉRA Proposition 2.1. Let C nd D be pir of closed nd convex subsets of reflexive Bnch spce X. IfC nd D hve the strong CHIP, nd t lest one of the following conditions holds: i) C D is bounded; ii) C D is flt; iii) C D is hlf-line, then the inf-convolution of the support functions of C nd D is exct. In other words, the converse of Moreu s theorem is vlid. Proof of Proposition 2.1. We need the following stndrd convex nlysis result. Lemm 2.2. Let C nd D be two closed nd convex subsets of loclly convex spce X, nd consider n element y of X expressed s the sum y = y 1 + y 2 of two norml vectors y 1 N C x) nd y 2 N D x), for some x C D. Then the inf-convolution of the support functionls is exct t y, tht is, 2.1) σ C σ D y) =σ C y 1 )+σ D y 2 )= x, y. Proof of Lemm 2.2. As y 1 ι C x) nd y 2 ι D x), reltion y = y 1 +y 2 implies tht y ι C D x). Thus nd hence 2.2) 2.3) σ C y 1 )= x, y 1,σ 2 y 2 )= x, y 2,σ C D y) = x, y, σ C D y) =σ C y 1 )+σ D y 2 ). Recll tht σ C D σ C σ D, which mens tht σ C D y) σ C σ D y). Finlly, use the definition of the inf-convolution nd the fct tht y = y 1 + y 2 to deduce tht 2.4) σ C σ D y) σ C y 1 )+σ D y 2 ). Reltion 2.1) follows from reltions 2.2), 2.3), nd 2.4). Let us now return to the proof of Proposition 2.1 nd consider tht cse i) holds; i.e., we suppose tht the pir C, D) hs the strong CHIP nd C D is bounded. As, in ddition, X is reflexive Bnch spce, it is esy to see tht, for every y X, there is n x C D such tht y ι C D x). The pir C, D) hs the strong CHIP, nd thus y = y 1 + y 2, for some y 1 ι C x) nd y 2 ι D x); we my therefore pply Lemm 2.2 nd deduce tht 2.5) σ C σ D y) =σ C y 1 )+σ D y 2 )= x, y. On one hnd, from reltion 2.5) we observe tht the Γ 0 X )-functionl σ C σ D is rel-vlued on X nd thus, s X is reflexive Bnch spce, is follows tht σ C σ D is continuous. Tking into ccount tht reltion 2.5) implies, on the other hnd, tht the infimum is lwys ttined in the expression of the inf-convolution, we conclude tht the inf-convolution of the support functions σ C nd σ D is exct. Cse ii). Let L be the closed subspce of X prllel to the flt C D tht is, C D = x 0 + L for every x 0 C D), nd fctorize X with respect to L. The quotient spce X/L, sy ˆX, is gin reflexive Bnch spce. Since x 0 +L C for every x 0 C, nd x 0 + L D for every x 0 D, it follows tht Ĉ nd ˆD, the
BOUNDARY HALF-STRIPS AND THE STRONG CHIP 5 quotients of the sets C nd D re closed, nd convex subsets of ˆX; moreover, it is strightforwrd to prove tht the pir Ĉ, ˆD) hs the strong CHIP if nd only if the sme holds for the pir C, D), nd tht the inf-convolution of the support functions of Ĉ nd ˆD is exct if nd only if the inf-convolution of the support functions of C nd D is exct. But the intersection between Ĉ nd ˆD reduces to singleton, nd cse ii) is proved by pplying the conclusion of cse i) to the pir Ĉ, ˆD). Cse iii). Set x 0 + R + x for the hlf-line C D. Obviously, when y X nd x, y 0wehvey ι C D x 0 ). Use the fct tht the pir C, D) hs the CHIP to deduce tht y = y 1 + y 2 for some y 1 ι C x 0 ) nd y 2 ι D x 0 ), together with Lemm 2.2, to infer tht 2.6) σ C σ D y) =σ C y 1 )+σ D y 2 )= x 0,y y X, x, y 0. In order to obtin similr reltion for the cse x, y > 0, note tht, for every z X such tht x, z > 0 it holds tht σ C z) =σ D z) =+. Moreover, the inequlity x, z + v > 0 mens tht t lest one of the inequlities x, z > 0 nd x, v > 0 holds. Combining these two fcts, we deduce tht which mens tht 2.7) σ C z)+σ D v) =+ z,v X such tht x, z + v > 0, Combining reltions 2.6) nd 2.7) yields σ C σ D y) =+ y X, x, y > 0. σ C σ D = ι {y X : x,y 0} + x 0,, which mens tht the inf-convolution of σ C nd σ D is the sum between the indictor function of hlf-spce nd liner nd continuous functionl, nd clerly belongs to Γ 0 X ). Use once more reltion 2.6) to see tht the infimum in the expression of the inf-convolution is chieved, nd conclude tht the inf-convolution of σ C nd σ D is exct. Apprently, Proposition 2.1 lists three completely disprte conditions, ech one being sufficient in its own wy for the vlidity of the converse of Moreu s theorem. The geometric notion of hlf-strip, tht is, convex hull of two prllel nd disjoint hlf-lines, llows us to spot common property of cses i), ii), nd iii) in Proposition 2.1. Theorem 2.3. Let C nd D two closed nd convex subsets of the Eucliden spce X, nd ssume tht the boundry of the set C does not contin ny hlf-strip. If the pir C, D) hs the strong CHIP, then the inf-convolution of σ C nd σ D is exct in other words, the converse of Moreu s theorem holds). Proof of Theorem 2.3. When the intersection C D meets the interior of C, we specify the well-known Moreu Rockfellr internl point condition see [9, Chp. 6, section 6.8]) to prove tht the inf-convolution of the support functionls is exct. If C D is prt of the boundry of C, use s the boundry of C does not contin ny hlf-strip the obvious fct tht the only closed nd convex subsets of n Eucliden spce which do not contin ny hlf-strip re the bounded sets, the hlflines, nd the lines, nd completely prove Theorem 2.3 by mking use of Proposition 2.1.
6 EMIL ERNST AND MICHEL THÉRA +A x 0 x 0 x 3 x 1 x 2 x +E 0 3 C y z O x Fig. 3.1. Closed nd convex set with boundry hlf-strip. 3. Convex sets with boundry hlf-strips. The following result completes the nlysis initited in Theorem 2.3. Theorem 3.1. Let C be closed nd convex subset of the Eucliden spce X; ssume moreover tht the boundry of C contins t lest one hlf-strip. Then there is closed nd convex set D such tht the pir C, D) hs the CHIP while the infconvolution between σ C nd σ D is not exct. In other words, the existence of t lest one boundry hlf-strip prevents the converse of Moreu s theorem from holding. 3.1. Construction nd properties of the set D. Our strtegy in this section is to construct the set D. The following esy result will be useful. It sys tht when X is Eucliden, every closed nd convex set with boundry hlf-strip my be contined within some hlf-spce such tht its boundry hlf-strip lies within the hyperplne which delimits this hlf-spce. Proposition 3.2. Let C be closed nd convex subset of n Eucliden spce X such tht its boundry contins hlf-strip. Then, there is n orthonorml bsis of X, B = {b 1,b 2,...,b n }, positive prmeter >0, nd x 0, n element of X, such tht where A is the hlf-strip defined s x 0 + A C x 0 + E 3, A = {x X :0 x b 1, 2 x b 2 2, 0=x b i i 3}, nd the hlf-spce E 3 is given by the reltion E 3 = {x X : x b 3 0}. Proof of Proposition 3.2. Let x 1, x 2, x in X be such tht x = 1 nd the hlfstrip spnned by the hlf-lines x 1 + R + x nd x 2 + R + x lies within the boundry of the set C. Assume if necessry fter chnging x 1 into x 2 ) tht x 2 x x 1 x, nd set x 3 = x 1 +[x 2 x 1 )x]x. Clerly, x 3 x 1 + R + x; s the hlf-lines x 1 + R + x nd x 2 + R + x re disjoint, it follows tht x 3 nd x 2 cnnot coincide. Set = x 3 x 2 4 nd y = x 3 x 2 4 = x 3 x 2 x 3 x 2.
Note tht BOUNDARY HALF-STRIPS AND THE STRONG CHIP 7 3.1) y x = 1 4 x 3 x x 2 x) = 1 4 x 1 x +x 2 x 1 )x x x x 2 x) =0. Finlly, put x 0 = x 3 + x 2 + x = 1 2 2 x 3 + x)+ 1 2 x 2 + x); s x 3 + x x 1 + R + x nd x 2 + x) x 2 + R + x), we deduce tht x 0 belongs to the hlf-strip spnned by the hlf-lines x 1 + R + x nd x 2 + R + x, nd thus belongs to the boundry of C. Since x 0 is boundry point of closed nd convex subset of n Eucliden spce, it is well known tht there is some liner mpping z X, z = 1, which chieves its mximum on C t x 0, 3.2) z x 0 z x x C. Apply reltion 3.2) for x = x 0 x = x3+x2 2 to deduce tht z x 0, nd then for x = x 0 + x = 1 2 x 3 +2x)+ 1 2 x 2 +2x) to obtin z x 0 nd therefore conclude tht 3.3) z x =0. Similrly, put x 0 x 2y = x 2 for x in reltion 3.2), nd using lso reltion 3.3), deduce tht z y 0. Finlly, putting x = x 0 x +2y = x 3 in reltion 3.2), nd lso tking into ccount reltion 3.3), we infer z y 0, tht is, 3.4) z y =0. Reltions 3.1), 3.3), nd 3.4) prove tht it is possible to complete the set {x, y, z} up to B = {b 1,b 2,...,b n }, n orthonorml bsis of X. The proof of Proposition 3.2 will be completed if we remrk tht the set x 0 + A is nothing but the hlf-strip spnned by the hlf-lines x 3 + x)+r + x nd x 2 + x)+ R + x, nd thus lies within the boundry of C, while reltion 3.2) implies tht C is prt of the hlf-spce x 0 + E 3. The bsis B, the prmeter, nd the element x 0 thus defined llow us to proceed to the construction of the set D. Let us first define the set F, F =P 1 + S) P 2 + T ), where P 1 P 2 re the sets bordered by two plne prbole: P 1 = {x X : x b 1 x b 2) 2 },x b i =0 i 3, 4 P 2 = {x X : x b 1 x b 2) 2 },x b i =0 i 3, 8 S is n orthogonl box in X: S = {x X : x b 1 0, 1 x b 2 1, x b 3 0},
8 EMIL ERNST AND MICHEL THÉRA b 3 b 1 O 1 S P 1 P 2 T b 2 Fig. 3.2. Sets needed in constructing the set D. T is closed nd convex subset of S: T = {x X : x b 1 0, 1 <x b 2 < 1, x b 3 2 x b 2 ) 2 } 1 2 x b 2 ) 2, nd, s customry, Z mens the polr set of some subset Z of X, Set now Z = {x X : x y 1 y Z}. D = x 0 + F = x 0 +P 1 + S) P 2 + T )). This definition grnts to the set F nd thus D) severl geometricl properties which re crucil for our purpose. Let us first notice tht F is contined in the hlf-spce E 1 = {x X : x b 1 0}; ccordingly, the hlf-line R + b 1 lies within F, nd thus the hlf-line x 0 + R + b 1 is prt of both sets C nd D = x 0 + F. It follows tht σ C x) =σ D x) =+ x X such tht x b 1 > 0. Let x X be such tht x b 1 =0;ify is such tht y b 1 0, then either y b 1 > 0or x y) b 1 > 0, so 3.5) σ C y)+σ D x y) =+ x X, x b 1 =0,y X, y b 1 0; hence for every element x X such tht x b 1 = 0 it results tht 3.6) σ C σ D x) = inf y X, y b 1=0 σ Cy)+σ D x y). The hyperplne L 1 = {x X : x b 1 =0} thus plys very importnt role in computing the inf-convolution of the support functions σ C nd σ D. The following lemm describes the intersection between the set F nd L 1 ; for convenience, we stte the result in terms of { } 1 γ F x) = inf s>0 s : sx F,
BOUNDARY HALF-STRIPS AND THE STRONG CHIP 9 b 3 b 1 O b 2 F T U L 1 L 1 b 2 λ b 3 Fig. 3.3. Intersection between F nd L 1. the guge function of the set F. Lemm 3.3. The set F is closed nd convex subset of the Eucliden spce X. Moreover, F L 1 = T L 1, nd thus 3.7) 3.8) nd 3.9) γ F x) x b 2 x L 1, γ F x) > 1 x L 1 such tht x b 2 =1, ) lim γ F λ + λb 3 =1. Proof of Lemm 3.3. Recll tht the sum Z 1 + Z 2 of two closed nd convex subsets Z 1 nd Z 2 of n Eucliden spce is lwys convex. This sum is moreover closed, provided tht Z 1 nd Z 2 do not contin two prllel hlf-lines see [12, Corollry 9.1.2] ). This is obviously the cse for the pirs of closed nd convex sets P 1 nd S, s well s P 2 n T, nd thus the sets P 1 + S nd P 2 + T re closed nd convex, nd the sme clerly holds lso for the set F, which is their intersection. It hs lredy been noticed tht ll the sets P 1, P 2, S, nd T ly within E 1 ; ccordingly, the sum of two elements x 1 nd x 2 from either P 1 nd S, orp 2 nd T, is contined within the delimiting hyperplne L 1 if nd only if both elements x 1 nd x 2 belong to L 1. In other words, P 1 + S) L 1 =P 1 L 1 )+S L 1 ), P 2 + T ) L 1 =P 2 L 1 )+T L 1 ), nd note tht P 1 L 1 = P 2 L 1 = {0} to deduce tht F L 1 =P 1 + S) L 1 ) P 2 + T ) L 1 ) =S L 1 ) T L 1 )=S T ) L 1.
10 EMIL ERNST AND MICHEL THÉRA Recll tht T S, nd deduce tht F L 1 = T L 1. This reltion my be used in order to compute the vlue of γ F x) for elements x L 1, since it obviously holds tht Use the fct tht to deduce tht γ F x) =γ F L1 x) =γ T L1 x) x L 1. T L 1 S L 1 M = {x X : x b 1 =0, 1 x b 2 1} γ T L1 x) γ M x) = x b 2 x L 1, tht is, reltion 3.7). In order to prove reltion 3.8), note tht, for every x T L 1 we hve x b 2 < 1. Accordingly, reltion x b 2 = 1 implies tht x/ T L 1. Let us now use [12, Corollry 9.7.1], which sys tht T L 1 = {y : γ T L1 y) 1}, nd deduce tht γ T L1 x) > 1. Finlly, if λ 1, stndrd computtion shows ) 4λ2 +1 1 2λ λb 3 T L 1 ; thus 3.10) 3.11) ) γ T L1 λb 2λ 3 4λ2 +1 1. Use reltion 3.8) for x = b 2 λb 3 ) to see tht 1 <γ T L1 λb 3 reltion 3.9) simply comes from reltions 3.10) nd 3.11). An importnt step in proving tht the pir of closed nd convex sets C, D) hs the strong CHIP is to determine their intersection C D. Lemm 3.4. It holds tht ) ; 3.12) C D = x 0 +F + Rb 3 ). Proof of Lemm 3.4. Use the fct tht R + b 3 ) T S to deduce tht R + b 3 ) F, nd thus tht F E 3 ). Accordingly, D x 0 + E 3 ), nd s C x 0 + E 3, we obtin tht where by L 3 we men In other words, C D x 0 + E 3 ) x 0 + E 3 )) = x 0 + L 3, L 3 = {x X : x b 3 =0}. 3.13) x 0 b 3 = x b 3 x C D.
BOUNDARY HALF-STRIPS AND THE STRONG CHIP 11 Consequently, 3.14) C D = C D x 0 + L 3 )) = C x 0 +F L 3 )). Recll see [2, Chpter 4, section 1, Corollry of Proposition 3]) tht, for every closed nd convex sets A nd B contining 0, it holds tht A B) = coa B ), where coa) denotes the closed convex hull of the set A. Thus, by using the bipolr theorem see [2, Chpter 4, section 1, Proposition 3]) pplied for the set F, nd the obvious fct tht L 3 = Rb 3, we deduce tht 3.15) F L 3 ) = cof L 3)=coF Rb 3 ). It is well known tht for every convex set A nd flt W, co A W )=co A+W ). Apply this reltion to the convex set F nd the one-dimensionl flt Rb 3 to prove tht co F Rb 3 )=co F + Rb 3 ); by virtue of reltion 3.15) it follows tht F L 3 ) = co F + Rb 3 ). Accordingly, F L 3 =F L 3 ) =co F + Rb 3 )) ; s the polr of ny set coincides with the polr of its closure, we hve 3.16) F L 3 =F + Rb 3 ). Let us prove tht the set F + Rb 3 ) lies within C. Indeed, fter n esy computtion it results tht 3.17) T + Rb 3 )=S + Rb 3 = N = {x X : x b 1 0, 1 x b 2 1} ; thus N F + Rb 3 ), which mens tht F + Rb 3 ) N. But s N = {x X :0 x b 1, x b 2, 0=x b i i 3}, we hve see Proposition 3.2) N A, nd thus 3.18) x 0 +F + Rb 3 ) x 0 + N ) x 0 + A) C. Reltion 3.12) follows now from reltions 3.14), 3.16), nd 3.18). It thus becomes necessry to determine the sum between the closed nd convex set F nd the line Rb 3. Lemm 3.5. It holds tht 3.19) {x P 1 + S) : x b 1 < 0} + Rb 3 F + Rb 3 P 1 + S)+Rb 3 ; ccordingly, 3.20) C D = x 0 +P 1 + S + Rb 3 ). Moreover, the guge functions γ F nd fulfill the following property: For every x X such tht x b 1 < 0, there is θx) 0 such tht 3.21) x) =γ F x θx)b 3 ).
12 EMIL ERNST AND MICHEL THÉRA b 3 P +S+Rb ) L 1 3 3 U b 1 O P 1 b 2 P 2 L 3 F Fig. 3.4. The sum between F nd the line Rb 3. Proof of Lemm 3.5. On the bsis of formul 3.17), we clim tht S + Rb 3 is closed nd convex set. Moreover, there re no prllel hlf-lines within P 1 nd S + Rb 3 ), so, using gin [12, Corollry 9.1.2], we deduce tht the set P 1 + S + Rb 3 is closed nd convex. Let us prove the second inclusion in 3.19). As F P 1 + S), it clerly follows tht 3.22) F + Rb 3 P 1 + S + Rb 3. To estblish the first inclusion in reltion 3.19), we prove nd use the fct tht, for every x P 1 + S such tht x b 1 < 0, there is λx) 0 such tht x λx)b 3 ) F. When 1 <x b 2 < 1, it is esy to see tht the vlue λx) =x b 3 + 2 x b 2 ) 2 1 2 x b 2 ) 2 does the job. Indeed, the element y =x b 1 )b 1 lies within both P 1 nd P 2, while z = x y λx)b 3 =x b 2 )b 2 2 x b 2 ) 2 1 2 x b 2 ) 2 b 3 + is obviously contined in T, nd thus in S. Accordingly, n x b i )b i x λx)b 3 = y +x y λx)b 3 ) P 1 + S) P 2 + T )=F. Let x P 1 + S) such tht x b 1 < 0 nd x b 2 1; to fix the ides, dmit tht x b 2 1. In order to define λx) in this cse, use the fct tht x cn be expressed s the sum x = y + z of two elements y nd z such tht y P 1 nd z S. As y P 1, it follows tht i=4 3.23) y b 1 y b 2) 2 ; 4
BOUNDARY HALF-STRIPS AND THE STRONG CHIP 13 since for every z S it holds tht z b 1 0, we deduce tht thus x b 1 y b 1.We my ccordingly infer from reltion 3.23) tht x b 1 y b 2) 2 3.24). 4 Use once more the fct tht z S, to conclude tht 1 z b 2 1. Recll tht x b 2 1, nd deduce tht y b 2 ) 2 =x b 2 z b 2 ) 2 x b 2 ) 1 2 3.25) ; from reltion 3.24) nd 3.25) it follows tht x b 1 x b 2 1 3.26). 4 Combine the fct tht x b 1 0 with reltion 3.26) nd deduce tht ) 2 x b 1 < x b 2 1 ; 8 ccordingly, for some prmeter α such tht 0 <α<1, we hve x b 1 < x b 2 α) 2 3.27). 8 We cn now define λx) s λx) = 2 α 2 1 2 α 2. Inequlity 3.27) proves tht the element x αb 2 n i=4 x b i)b i ) belongs to the set P 2 ; s obviously we deduce tht αb 2 2 α 2 1 2 α 2 b 3 + ) 2 n x b i )b i T, i=4 3.28) x λx)b 3 = x αb 2 P 2 + T. ) ) n x b i )b i + αb 2 2 α 2 n 1 2 α 2 b 3 + x b i )b i i=4 i=4 Remrk tht the cse x b 2 1 is similr to the cse x b 2 1. Indeed, when x b 2 1, one hs y b 2 ) 2 x b 2 + 1 ) 2 insted of reltion 3.25). The prmeter α now lies between 1 nd 0, nd fulfills x b 1 < x b 2 + α) 2, 8
14 EMIL ERNST AND MICHEL THÉRA nd not reltion 3.26). As in the cse x b 2 1, x is the sum of two elements: one in P 2, the other in T. However, when x b 2 1, the element belonging to P 2 is x + αb 2 n i=4 x b i)b i ), nd the one lying in T is αb 2 2 α 2 1 2 α b 2 3 + n i=4 x b i)b i. Noticing tht S + R + b 3 )=S, we hve P 1 + S + R + b 3 )=P 1 + S). Thus, s x P 1 + S, we deduce tht 3.29) x λb 3 P 1 + S λ 0; from 3.28) nd 3.29) it follows tht x λx)b 3 ) P 1 + S) P 2 + T )=F, nd therefore for every x P 1 + S such tht x b 1 < 0, there is λx) 0 such tht x λx)b 3 F. Use this observtion to prove tht {x P 1 + S : x b 1 < 0} + Rb 3 F + Rb 3, which, together with reltion 3.22), yields reltion 3.19). Reltion 3.19) implies tht the set P 1 + S + Rb 3 is the closure of the set F + Rb 3. Reclling tht the polr of ny set coincides with the polr of its closure, we deduce tht 3.30) F + Rb 3 ) =P 1 + S + Rb 3 ), nd reltion 3.20) follows from formuls 3.12) nd 3.30). It remins to prove reltion 3.21). To begin with, notice tht, from reltion 3.19) it follows tht F P 1 + S + Rb 3, nd thus 3.31) γ F. Let us first prove tht x) is rel-vlued for every x X such tht x b 1 < 0. In this respect, note tht from reltion 3.17) it results tht {x X : x b 2 = x b 3 =0} S + Rb 3, nd thus tht 3.32) P 1 + {x X : x b 1 = x b 2 =0} P 1 + S + Rb 3 ). On the other hnd, the set P 1 + {x X : x b 1 = x b 2 =0} contins ll the elements x X such tht x b 1 x )2 4.As ) ) 2 4x b ) 1 x b 2 ) 2 x b 1 = 4x b 1) 2 4x b1 x b 2 ) 2 = x b 2) x b 2 2, 4 which mens tht 4x b 1 x b 2 ) 2 x P 1 + {x X : x b 1 = x b 2 =0}. Combine the previous reltion with formul 3.32) to deduce tht x) x b 2) 2 4x b 1.
BOUNDARY HALF-STRIPS AND THE STRONG CHIP 15 Consequently, for every x X such tht x b 1 < 0wehve x) < +. Let us first consider the cse when x) = 0, tht is, when x belongs to ry completely contined in P 1 + S + Rb 3. Tking into ccount the definitions of the sets P 1 nd S, wehve similrly, [ x) =0] [x b 1 0 nd x b 2 =0]; [γ F x) =0] [x b 1 0,x b 2 = 0 nd x b 3 0]. In this cse, θx) = x b3+ x b3 2 obviously does the job. Let us now turn to the cse x) > 0 nd write tht x x) P 1 + S + Rb 3. We deduce tht there is λx) R such tht x x) λx)b 3 P 1 + S. Accordingly, which mens tht 3.33) x x) λx)b 3 γ F x x) ) λ x x) λx)b 3 ) b 3 F, )) ) x λx)+λ x) λx)b 3 b 3 x). Reltions 3.31), 3.33) nd the obvious fct tht prove reltion 3.21) with x) = x + νb 3 ) θx) = x) λx)+λ ν R x x) λx)b 3 )), completing in this wy the proof of Lemm 3.5. 3.2. The min result. We clim tht the pir of closed nd convex sets C nd D hs the strong CHIP. Proposition 3.6. The pir of closed nd convex subsets C nd D of the Eucliden spce X hs the strong CHIP. Proof of Proposition 3.6. Let x 1 C D nd y ι C D x 1 ), y 0. Our im is to express y s the sum of two elements y 1 nd y 2 from ι C x 1 ) nd ι D x 1 ). Let us first remrk tht, since x 0 + A) C x 0 + E 3 ) see Proposition 3.2), it follows tht 3.34) R + b 3 ι C x)
16 EMIL ERNST AND MICHEL THÉRA for every x x 0 + A, in prticulr for every x C D see Lemm 3.4 nd reltion 3.18)). Similrly, we deduce tht 3.35) R + b 3 ) ι D x) for every x C D. The flt {x X : x b i =0 1 i 3} obviously lies within T, nd thus in S, hence in F.ThusD is contined within the flt x 0 + {x X : x b i = 0 i 4}, nd we deduce tht 3.36) {x X : x b 1 = x b 2 = x b 3 =0} ι D x) x D. From reltions 3.34), 3.35), nd 3.36) it follows tht 3.37) {x X : x b 1 = x b 2 =0} ι C x)+ ι D x)) x C D). We ddress first the cse when y b 1 = 0. A stndrd computtion shows tht 3.38) P 1 + S + Rb 3 ) = {x X : x b 1 x b 2) 2 } x b 2, <x b 2 <, x b i =0 i 3. From the previous reltion it follows tht the set P 1 + S + Rb 3 ) is contined within the plne spnned by b 1 nd b 2. For every y X such tht y b 1 = 0 it follows tht 3.39) x y =x b 2 )y b 2 ) x P 1 + S + Rb 3 ). The elements x 1 nd x 0 re both in C D; in view of reltion 3.20) it ppers tht x 1 x 0 ) P 1 + S + Rb 3 ). From reltion 3.38) it follows tht x 1 x 0 ) b 2 <. Set α = + x 1 x 0 ) b 2,z 1 = α2 2 α b 1 αb 2,z 2 = α2 α b 1 + αb 2 ; thus z 1,z 2 P 1 + S + Rb 3 ) nd z 1 b 2 < x 1 x 0 ) b 2 <z 2 b 2. Recll tht, s y ι C D x 1 ), the liner functionl X x x y) R chieves its mximum on C D t x 1. Thus, on one hnd, tht is, in view of reltion 3.39), x 0 + z 1 ) y x 1 y, z 1 b 2 )y b 2 ) x 1 x 0 ) b 2 )y b 2 ); combine this reltion with the fct tht z 1 b 2 y b 2 0. On the other hnd, < x 1 x 0 ) b 2, nd deduce tht x 0 + z 2 ) y x 1 y, tht is, once more by virtue of reltion 3.39), z 2 b 2 )y b 2 ) x 1 x 0 ) b 2 )y b 2 ); in ddition, s x 1 x 0 ) b 2 <z 2 b 2,wegety b 2 0.
BOUNDARY HALF-STRIPS AND THE STRONG CHIP 17 We my thus conclude tht, when y b 1 = 0, it results tht y b 2 = 0, nd formul 3.37) proves tht y is the sum of two elements from ι C x 1 ) nd ι D x 1 ). Consider now the cse when y b 1 0, which, tking into ccount the fct tht the hlf-line x 0 + R + b 1 ) is contined s lredy remrked) within C D, mounts to sying tht y b 1 < 0. It is well known [12, Theorem 14.5]) tht, for every closed nd convex set Z contining 0 it holds σ Z = γ Z. Use this reltion for the set P 1 + S + Rb 3 ) to obtin σ P1+S+Rb 3) = γ P 1+S+Rb 3 ; s see 3.20) P 1 + S + Rb 3 ) =C D) x 0, it follows tht 3.40) σ C D) x0 =. Similrly, 3.41) σ D x0 = γ F. From reltions 3.21), 3.40), nd 3.41) it follows tht there is some θy) 0 such tht thus 3.42) 3.43) tht is, σ C D) x0 y) =σ D x0 y θy)b 3 ); σ C D y) =σ D y θy)b 3 )+θy)x 0 b 3. As y ι C D x 1 ), it results tht σ C D y) =x 1 y; use reltion 3.42) to see tht σ D y θy)b 3 )+θy)x 0 b 3 = x 1 y. Reltion 3.13) reds x 0 b 3 = x 1 b 3. Equlity 3.43) my thus be stted s σ D y θy)b 3 )+θy)x 1 b 3 = x 1 y, σ D y θy)b 3 )=x 1 y θy)b 3 ). This mens tht y θy)b 3 ) ι D x 1 ). Recll see reltion 3.34)) tht λb 3 ι C x 1 ) for every λ 0, nd express y s y = θy)b 3 +y θy)b 3 ), tht is, the sum of n element from ι C x 1 ) nd the sum of nother element from ι D x 1 ). We finlly clim tht the inf-convolution of the support functionls σ C nd σ D is not exct t b 2, fct which completes the proof of Theorem 3.1. Proposition 3.7. It holds tht 3.44) while σ C σ D ) = x 0 b 2 +1, 3.45) σ C y)+σ D z) > x 0 b 2 +1 y + z = b 2.
18 EMIL ERNST AND MICHEL THÉRA Proof of Proposition 3.7. Use the fct tht x 0 +2b 2 ) nd x 0 2b 2 ) both belong to x 0 + A, nd thus to C, to deduce tht 3.46) σ C x) mx x 0 +2b 2 ) x, x 0 2b 2 ) x) =x 0 x +2 x b 2. From reltion 3.7) it follows tht γ F x) x b 2 x L 1. Reltion 3.41) reds σ D x0 = γ F ; hence, it results tht 3.47) σ D x) =x 0 x + γ F x). It follows tht 3.48) σ D x) x 0 x + x b 2 x L 1. From reltions 3.46) nd 3.48) it results tht ) 3.49) σ C x)+σ D x ) x 0 x +2 x b 2 + x 0 x + 3.50) ) x b 2 x 0 b 2 +1+ x b 2 x X, x b 1 =0. By tking into ccount reltions 3.6) nd 3.49) we prove tht σ C σ D As x 0 C x 0 + E 3 ), it follows tht Use reltion 3.47) to deduce tht ) σ C λb 3 )+σ D λb 3 ) x 0 b 2 +1. σ C λb 3 )=λx 0 b 3 λ 0. = λx 0 b 3 + x 0 b 2 λx 0 b 3 + γ F λb 3 ). From the previous equlity, together with reltion 3.9), it yields tht )) lim σ C λb 3 )+σ D λ λb 3 = x 0 b 2 +1, which, combined with inequlity 3.50), proves reltion 3.44). Finlly, let x L 1 be such tht x b 2 = 0. Then see reltion 3.8)) γ F x ) > 1,
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