| Summary: | <p>Let <em>S</em> be a set of <em>n</em> points in <strong>R</strong><sup>d</sup>. A Steiner convex partition is a tiling of conv(<em>S</em>) with empty convex bodies. For every integer <em>d</em>, we show that <em>S</em> admits a Steiner convex partition with at most ⌈(<em>n</em>-1)/<em>d</em>⌉ 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 <em>d</em>≥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.</p><p>Establishing a tight lower bound for the maximum volume of a tile in a Steiner convex partition of any <em>n</em> 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/<em>n</em>). Here we give a (1-\epsilon)-approximation algorithm for computing the maximum volume of an empty convex body amidst <em>n</em> given points in the <em>d</em>-dimensional unit box [0,1]<sup>d</sup>.</p>
|
|---|
