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Thieves Can Make Sandwiches Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Thieves can make sandwiches Pavle V. M. Blagojevi´cand Pablo Sober´on Dedicated to Imre B´ar´any on the occasion of his 70th birthday Abstract We prove a common generalization of the Ham Sandwich theorem and Alon's Necklace Splitting theorem. Our main results show the existence of fair distributions of m measures in Rd among r thieves using roughly mr=d convex pieces, even in the cases when m is larger than the dimension. The main proof relies on a construction of a geometric realization of the topological join of two spaces of partitions of Rd into convex parts, and the computation of the Fadell-Husseini ideal valued index of the resulting spaces. 1. Introduction Measure partition problems are classical, significant and challenging questions of Discrete Geometry. Typically easy to state, they hide a connection to various advanced methods of Algebraic Topology. In the usual setting, we are presented with a family of measures in a geometric space and a set of rules to partition the space into subsets, and we are asked if there is a partition of this type which splits each measure evenly. In this paper we consider convex partitions of the Euclidean space Rd. More precisely, an ordered d d d collection of n closed subsets K = (K1;:::;Kn) of R is a partition of R if it is a covering R = K1 [···[ Kn, all the interiors int(K1);:::; int(Kn) are non-empty, and int(Ki) \ int(Kj) = ; for all d 1 ≤ i < j ≤ n. A partition K = (K1;:::;Kn) is a convex partition of R if each element Ki of the partition is convex. Furthermore, for an integer r ≥ 1 an r-labeled (convex) partition of Rd is an ordered pair (K; `) d where K = (K1;:::;Kn) is a (convex) partition of R , and `:[n] −! [r] is an arbitrary function. We use the notation [n] := f1; : : : ; ng. Throughout this paper all measures in Rd are assumed to be probability measures that are absolutely continuous with respect to the Lebesgue measure. This in particular means that the overlapping boundary S 1≤i<j≤n Ki \ Kj of the elements of a partition always has measure zero. The quintessential measure partitioning result is the Ham Sandwich theorem, which was conjectured by Steinhaus and proved subsequently by Banach in 1938 (consult for example [BZ04]). The Ham Sandwich theorem is one of the most widely known consequences of the Borsuk{Ulam theorem. Theorem (Ham Sandwich theorem). Let d ≥ 1 be an integer. For any collection of d measures d µ1; : : : ; µd in R , there exists an affine hyperplane H that simultaneously splits them into halves. Namely, we have + − µi(H ) = µi(H ) for all 1 ≤ i ≤ d, where H+ and H− denote closed half-spaces determined by H. arXiv:1706.03640v2 [math.CO] 28 Sep 2017 The reason for its name is an illustration where each of the measures is thought of as a different ingredient floating in Rd. The goal is to make two sandwiches with equal amount of each ingredient by cutting Rd with a single hyperplane slice. Furthermore, if more people want their sandwich and they all fancy convex shapes, it has been shown by different groups of authors [Sob12][KHA14][BZ14] 2000 Mathematics Subject Classification 52A99 , 52A37, 55S91 (primary), 05A18 (secondary). Please refer to http://www.ams.org/msc/ for a list of codes. Pavle V. M. Blagojevi´creceived funding from DFG via the Collaborative Research Center TRR 109 \Discretization in Geometry and Dynamics", and the grant ON 174008 of the Serbian Ministry of Education and Science. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while Pavle V. M. Blagojevi´cwas in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall of 2017. Page 2 of 14 PAVLE V. M. BLAGOJEVIC´ AND PABLO SOBERON´ d that for any collection of d measures µ1; : : : ; µd in R and any integer r ≥ 1 there is a convex partition d K = (K1;:::;Kr) of R into r parts that simultaneously split each measure into r parts of equal size. Namely, µi(K1) = ··· = µi(Kr) for all 1 ≤ i ≤ d. The second classic and appealing measure partition result, and again a consequence of the Borsuk{Ulam theorem, is the following \Necklace Splitting" theorem of Hobby and Rice [HR65]. Theorem (Necklace Splitting theorem). Let m ≥ 1 be an integer. Then for any collection of m measures µ1; : : : ; µm in R, there exists a 2-labeled convex partition (K; `) of R into a collection of m + 1 intervals K = (K1;:::;Km+1) with `:[m + 1] −! [2] such that for every 1 ≤ j ≤ m: [ [ 1 µj Ki = µj Ki = 2 : i2`−1(1) i2`−1(2) The discrete version of this theorem, where the measures are the counting measures of finite sets of points, was first proved by Goldberg and West [GW85], and then by Alon and West [AW86]. The common illustration of this theorem is as follows. Two thieves steal an open necklace with m types of pearls, knowing that there is an even number of pearls of each kind. They will cut the necklace into pieces and distribute those among themselves, so that each receives half of each kind of pearl. The result above shows that this can always be achieved with m cuts, regardless of the order of the pearls. The function ` is simply telling us who gets each part. The version with arbitrary number r of thieves was given by Alon in [Alo87], where (r − 1)m cuts are shown to be sufficient. The number of cuts cannot be improved. Extensions of this result with additional combinatorial conditions on the distribution of the necklace appear in [AFP+17]. Our main goal in this manuscript is to present a common generalization of the Ham Sandwich theorem and the Necklace Splitting theorem. In other words, given more than d ingredients in Rd, we should be able to find a fair distribution among r hungry persons if we are willing to split Rd into more than r parts. Alternatively, if r thieves steal a high-dimensional necklace, they should be able to distribute it among themselves by splitting it into very few convex parts. High-dimensional versions of the Necklace Splitting theorem were given by de Longueville and Zivaljevi´cˇ [DLZ08ˇ ], and by Karasev, Rold´an-Pensado and Sober´on[KRPS16]. In [DLZ08ˇ ], the authors proved an analogous result for r thieves and m measures in Rd, where the partitions are made using (r − 1)m hyperplanes each of whose directions is fixed in advance and must be orthogonal to a vector of the canonical basis of Rd. The downside of this type of partitions is that there may be an extremely large number of pieces to distribute. Figure 1. An iterated partition of the plane by successive hyperplane cuts One way to address this issue is to consider iterated hyperplane partitions. A partition of Rd into convex parts is an iterated hyperplane partition if it can be made out of Rd by successively partitioning each convex part, from a previous partition, with a hyperplane (that means each new hyperplane only cuts one of the existing convex parts, see Figure 1). In [KRPS16] the authors showed that for r thieves and m measures in Rd, there is a fair distribution of each measure among the thieves using an iterated THIEVES CAN MAKE SANDWICHES Page 3 of 14 hyperplane partition that has (r − 1)m hyperplane cuts, whose directions are fixed in advance, as long as r is a prime power. This has the advantage that the total number of parts is (r − 1)m + 1. In both results, there is little to gain from the increasing dimension. This is a consequence of fixing the directions of the cutting hyperplanes. Thus, it is natural to wonder what can be gained if the fixed directions restriction is disregarded. In this situation we distinguish three different types of labeled partitions. The first type of partitions are labeled partitions of Rd into n convex parts without any additional requirements. For the second type of partitions we consider iterated convex partitions of Rd that in the case when r = 2 coincide with iterated hyperplane partitions. Intuitively, these are formed by splitting Rd using iterated hyperplane cuts, and then splitting each remaining region with a power diagram into r parts. Because of its complexity, the formal definitions of power diagrams and iterated convex partitions are postponed for the next section, Definition 4. Finally, the third type of partitions are those made by iterated hyperplane cuts. Definition 1. Let n ≥ 1, r ≥ 1 and d ≥ 1 be integers. (1) Let M = M(n; r; d) be the largest integer such that for any collection of M measures µ1; : : : ; µM d d in R there exists an r-labeled convex partition (K; `) of R into n parts K = (K1;:::;Kn) with `:[n] −! [r] having the property that for all 1 ≤ j ≤ M and all 1 ≤ s ≤ r: [ 1 µj Ki = r : i2`−1(s) Every such labeled convex partition into n parts is called a fair distribution between the thieves. (2) If n is a multiple of r, let M 0 = M 0(n; r; d) be the largest integer such that for any collection of M 0 d d measures µ1; : : : ; µM 0 in R there exists an r-labeled iterated partition (K; `) of R into n convex 0 parts K = (K1;:::;Kn) with `:[n] −! [r] so that for all 1 ≤ j ≤ M and all 1 ≤ s ≤ r: [ 1 µj Ki = r : i2`−1(s) Every such r-labeled convex partition into n parts is called a fair iterated distribution between the thieves.
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