Minimum Convex Partitions and Maximum Empty Polytopes

TL;DR

This paper establishes optimal bounds for Steiner convex partitions of point sets in various dimensions, introduces approximation algorithms for minimal partitions and maximum volume tiles, and connects these problems to longstanding conjectures in geometry.

Contribution

It provides tight bounds for Steiner convex partitions, the first constant-factor approximation for minimal partitions in the plane, and a $(1- ext{epsilon})$-approximation for maximum volume empty convex bodies.

Findings

Bound of $ ceil (n-1)/d ceil$ tiles for Steiner convex partitions.

First constant-factor approximation algorithm for minimal partitions in the plane.

A $(1- ext{epsilon})$-approximation algorithm for maximum volume empty convex bodies.

Abstract

Let $S$ be a set of $n$ points in $\mathbb{R}^d$. A Steiner convex partition is a tiling of ${\rm conv}(S)$ with empty convex bodies. For every integer $d$, we show that $S$ admits a Steiner convex partition with at most $\lceil (n-1)/d\rceil$ 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 $d\geq 3$. 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 $\omega(1/n)$. Here we give a $(1-\varepsilon)$-approximation algorithm for computing the maximum volume of an empty convex body amidst $n$ given points in the $d$-dimensional unit box $[0,1]^d$.