Complements of finite unions of convex sets

TL;DR

This paper systematically studies the complements of finite unions of convex sets, analyzing isolated points, Betti numbers, and non-convexity, and provides bounds on the structure and coverings of such sets.

Contribution

It introduces new bounds on isolated points related to Betti numbers and non-convexity, and characterizes minimal coverings of these complements by flats of varying dimensions.

Findings

Upper bounds on isolated points, sharp for n=3

Bounds on the size of minimal flat covers

Structural characterization of minimal covers

Abstract

Finite unions of convex sets are a central object of study in discrete and computational geometry. In this paper we initiate a systematic study of complements of such unions -- i.e., sets of the form $S=\mathbb{R}^d \setminus (\cup_{i=1}^n K_i)$, where $K_i$ are convex sets. In the first part of the paper we study isolated points in $S$, whose number is related to the Betti numbers of $\cup_{i=1}^n K_i$ and to its non-convexity properties. We obtain upper bounds on the number of such points, which are sharp for $n=3$ and significantly improve previous bounds of Lawrence and Morris (2009) for all $n \ll \frac{2^d}{d}$. In the second part of the paper we study coverings of $S$ by well-behaved sets. We show that $S$ can be covered by at most $g(d,n)$ flats of different dimensions, in such a way that each $x \in S$ is covered by a flat whose dimension equals the `local dimension' of $S$ in the neighborhood of $x$. Furthermore, we determine the structure of a minimum cover that satisfies this property. Then, we study quantitative aspects of this minimum cover and obtain sharp upper bounds on its size in various settings.