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Combinatorics and Convexity 1 Convex Sets in General Combinatorics and Convexity Gil Kalai Hebrew University Jerusalem Connections b etween Euclidean convex geometry and combinatorics go back to Euler Cauchy Minkowski and Steinitz The theory was advanced greatly since the s and was inuenced by the discovery of the simplex algorithm the connections with extremal combinatorics the intro duction of metho ds from commutative algebra and the relations with complexity theory The rst part of this pap er deals with convexity in general and the second part deals with the combinatorics of convex p olytop es There are many excellent surveys and collections of op en problems I try to discuss several sp ecic topics and to zo om in on issues which I am more familiar with Convex sets in general Covering Packing and Tiling d Borsuk conjectured that every b ounded set in R can b e covered by d sets of smaller diameter Kahn and Kalai showed that Borsuks conjecture is very false in high dimensions Here is the dispro of of Borsuks conjecture Let f d b e the smallest integer such that every d b ounded set in R can b e covered by f d sets of smaller diameter For a b ounded metric space X let bX b e the minimum numb er of sets of smaller diameter needed to cover X Consider d d P the space of lines through the origin in R where the metric is given by the angle b etween d two lines The diameter of P is and the distance b etween two lines is i they are orthogonal Let d p p a prime Frankl and Wilson see also proved that there are d d d d at most vectors in f g such that no two are orthogonal This yields bP d d since if P is covered by t sets of smaller diameter each such set contains at most of the d d d lines spanned by the vectors in f g But there are such lines and therefore t 2 d d d Now emb ed P into R by the map x x x where x is a vector of norm in R Note that x x y y x y Therefore the order relation b etween distances is preserved and p d d the image of P is the required counterexample This example gives f d for suciently large d Betke Henk and Wills proved for suciently high dimensions Fejes Toths sausage conjecture They showed that the minimum volume of the convex hull of n nonoverlapping congruent balls in d R is attained when the centers are on a line d Keller conjectured that in every tiling of R by cub es there are two cub es which share a complete facet Lagarias and Shor showed this to b e false for d They used a reduction to a purely combinatorial problem which was found by Corradi and Szabo 1 If x x x x and y y y y you can regard x y as the d k matrix whose i j entry is x y 1 2 d 1 2 k i j Some problems There are many problems on packing covering and tiling and the most famous are p erhaps the d sphere packing problem in R and the asymptotic sphere packing problem in R There are several op en problems around Borsuks problem What is the asymptotic b ehavior of f d What is the n situation in low dimensions What is the b ehavior of bP Witsenhausen conjectured see that if A is a subset of the unit sphere without two orthogonal vectors then v ol A v where p n n v is the volume of a spherical cap of radius This would imply that bP o Perhaps the algebraic metho ds used for the FranklWilson theorem can b e of help p d Schramm proved an upp er b ound f d sd o He showed that every set of constant width can b e covered by sd smaller homothets Bourgain and Lindenstrauss proved the same b ound by covering every b oundede set by sd balls of the same diameter Danzer already showed that exp onential numb er of balls is sometimes necessary In his pro of Schramm related the value of f d with another classical problem in convexity that of nding or estimating the minimal volume of Euclidean and more generally spherical sets of constant width d It is not known if there are sets of constant width in R whose volume is exponential ly smaller than the volume of a ball of radius Perhaps the following series of examples suggested by d Schramm K R will do but we do not know to compute or estimate their volumes K d and K is obtained as follows Consider K as sitting in the hyp erplane given by x in d d d d R Now take K A B where A is the set of all p oints z with x such that d d d d d the ball of radius around z contains K and B is the set of all p oints z with x which d d d b elong to every ball of radius which contains K Schramm also conjectured that the minimal d volume of a spherical set of constant width is obtained for an orthant Finally what is the minimal diameter d such that the unit nball can b e covered by n sets n of diameter d It is known that O log nn d O n see Hadwiger conjectured n n that the upp er b ound which corresp onds to the standard symmetric decomp osition of the ball to n regions is the truth Perhaps also here the natural conjecture is false Hellytyp e theorems Tverb ergs theorem Sarkaria found a striking simple pro of of the following theorem of Tverb erg d Every d r points in R can be partitioned into r parts such that the convex hul ls of these parts have nonempty intersection d He used the following result of Barany Let A A A be sets in R such that x d conv A for every i Then it is possible to choose a A such that x conv a a a i i i d To prove this consider the minimal distance t b etween x and such conv a a a and show d that if t one of the a s can b e replaced to decrease t i d Now consider m d r p oints a a a in R and regard them as p oints in m d V R whose sum of co ordinates is Sarkarias idea was to consider the tensor pro duct V W where W is a r dimensional space spanned by r vectors w w w whose sum is zero r Next dene m dimV U sets in V U as follows A fa w a w a w g i i i i r Note that is in the convex hull of each A and by Baranys theorem conv fa w a i i 1 w a w g for some choices of i i i The required partition of the p oints is given i m i m m 2 P by fa i j g j r To see this write a w where the co ecients j k k k k i k k P are nonnegative and sum to Deduce that the vectors v a j r are all equal j k k k j P and so are the scalars k j k j There are many b eautiful problems and results concerning Tverb ergs theorem see Top o logical versions were found for the case where r is a prime and were extended to derive colored versions of Tverb ergs theorem Sierksma conjectured see that the numb er of Tverb erg d d r partitions is at least r For a nite set A in R let f A r maxfdim conv g where i i the maximum is taken over all partitions of A r P jAj Conjecture f A r Note dim r This extension of Tverb ergs theorem was proved by Kadari for planar sets The HadwigerDebrunner Piercing Conjecture Alon and Kleitman proved the HadwigerDebrunner Piercing Conjecture For every d and every p d there is a c cp d such that the fol lowing holds For d every family H of compact convex sets in R in which any set of p members of the family contains d a subset of cardinality d with a nonempty intersection there is a set of c points in R that intersects each member of H Hellys theorem asserts that cd d and it is not dicult to see that cp p We describ e the pro of for the rst typical case d p We are given a family of n planar convex sets and out of every four sets in the family we can nail three with a p oint We want to nail the entire family with a xed numb er of p oints The rst step is to show that there is a way to nail a constant fraction indep endent from n of the sets with one p oint This follows from a fractional Helly theorem of Katcalski and Liu A more sophisticated use of the Katcalski Liu theorem shows that for every assignment of nonnegative weights to the sets in the family we can nail with one p oint sets representing a constant prop ortion of the entire weight Using linear programming duality Alon and Kleitman pro ceeded to show that there is a collection Y of p oints their numb er may dep end on n such that every set in the family is nailed by a constant fraction of the p oints in Y The nal step replacing Y with a set of b ounded cardinality which meets all the sets in the family is done using the theorems of Barany and Tverb erg mentioned ab ove Convex p olytop es Polytop es spheres and Steinitz theorem Convex p olytop es are among the most ancient mathematical ob jects of study The combinatorial theory of p olytop es is the study of their facestructure and in particular their face numb ers There is also a develop ed metric theory of p olytop es problems concerning volume width sections pro jections etc and arithmetic theory lattice p oints in p olytop es These three asp ects of convex p olytop es are related and some of the algebraic to ols mentioned b elow are relevant to all of them A convex ddimensional p olytop e briey a dp olytop e is the convex hull of a nite set of d p oints which anely span R A nontrivial face of a dp olytop e P is the intersection of P with a supp orting hyp erplane The empty set and P itself are regarded as trivial faces faces are called vertices faces are called edges and d faces are called facets The set of faces of a p olytop e is a graded lattice Two p olytop es P and Q are combinatorial ly isomorphic if there is an order preserving bijection b etween their face lattices P and Q are dual if there is an order reversing bijection b etween their face lattices Simplicial p olytop es are p olytop es all whose prop er faces are simplices
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