Minimum Convex Partitions and Maximum Empty Polytopes

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 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 .