Cutting a Part From Many Measures

Abstract

Holmsen, Kynčl and Valculescu recently conjectured that if a finite set X with n points in R d that is colored by m different colors can be partitioned into n subsets of points each, such that each subset contains points of at least d different colors, then there exists such a partition of X with the additional property that the convex hulls of the n subsets are pairwise disjoint.We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least c different colors, where we also allow c to be greater than d. Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from c different colors. For example, when n ≥ 2, d ≥ 2, c ≥ d with m ≥ n(c − d) + d are integers, and µ 1 , . . . , µm are m positive finite absolutely continuous measures on R d , we prove that there exists a partition of R d into n convex pieces which equiparts the measures µ 1 , . . . , µ d−1 , and in addition every piece of the partition has positive measure with respect to at least c of the measures µ 1 , . . . , µm. 1 arXiv:1710.05118v3 [math.CO] 24 Jun 2019 2 BLAGOJEVIĆ, PALIĆ, SOBERÓN, AND ZIEGLERLet m ≥ 1, n ≥ 1 , c ≥ 1 and d ≥ 1 be integers, and let M = (µ 1 , . . . , µ m ) be a collection of m finite absolutely continuous measures in R d . Moreover, assume that µ j (R d ) > 0, for every 1 ≤ j ≤ m. For us a measure is an absolutely continuous measure if it is absolutely continuous with respect to the standard Lebesgue measure.We are interested in the existence of a convex partition (C 1 , . . . , C n ) of R d with the property that each set C i contains a positive amount of at least c of the measures, that isfor every 1 ≤ i ≤ n. In the case when the measures are given by finite point sets, we say that a point set X ⊆ R d is in general position if no d + 1 points from X lie in an affine hyperplane in R d . For the point set measures in general position Holmsen, Kynčl and Valculescu proposed the following natural conjecture [12, Con. 3].Conjecture 1.2 (Holmsen, Kynčl, Valculescu, 2017). Let d ≥ 2, ≥ 2, m ≥ 2 and n ≥ 1 be integers with m ≥ d and ≥ d. Consider a set X ⊆ R d of n points in general position that is colored with at least m different colors. If there exists a partition of the set X into n subsets of size such that each subset contains points colored by at least d colors, then there exists such a partition of X that in addition has the property that the convex hulls of the n subsets are pairwise disjoint.The conjecture was settled for d = 2 in the same paper by Holmsen, Kynčl and Valculescu [12]. On the other hand, if instead of finite collections of points one considers finite positive absolutely continuous measures in R d , Soberón [16] gave a positive answer on splitting d measures in R d into convex pieces such that each piece has positive measure with respect to each of the measures. Moreover, he proved existence of convex partitions that equipart all measures. A discretization of Soberón's result by Blagojević, Rote, Steinmeyer and Ziegler [5] gave...