Minimum Convex Partitions and Maximum Empty Polytopes

Abstract

Let S be a set of n points in R d . A Steiner convex partition is a tiling of conv(S) with empty convex bodies. For every integer d, we show that S admits a Steiner convex partition with at most ⌈(n − 1)/d⌉ tiles. This bound is the best possible for points in general position in the plane, and it is best possible apart from constant factors in every fixed dimension d ≥ 3. We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position.Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any n points in the unit cube is equivalent to a famous problem of Danzer and Rogers. It is conjectured that the volume of the largest tile is ω(1/n). Here we give a (1 − ε)-approximation algorithm for computing the maximum volume of an empty convex body amidst n given points in the d-dimensional unit box [0,1] d .

Peeling Digital Potatoes

“…Note that the input of the continuous and digital problems is intrinsically different, hence we cannot compare the complexity of the two problems. Related continuous problems have been studied, such as the maximum volume of an empty convex body amidst n points [18], or the optimal island problem [6,22], in which we are given two sets S p , S n ⊂ R 2 , and the goal is to determine that largest subset K ⊆ S p such that conv(K) ∩ S n = ∅.…”

Peeling Potatoes Near-Optimally in Near-Linear Time

“…Dumitrescu, Har-Peled and Tóth [21] consider the following problem: given a unit square Q and a set X of points inside Q, find a maximum-area convex body inside Q that does not have any point of X in its interior. This is an instance of the potato peeling problem for polygons with holes.…”