Minimum convex partitions and maximum empty polytopes

Abstract

Let S be a set of n points in Rd. 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 the 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-\epsilon)-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.