Complements of finite unions of convex sets

2025-08-26

AI Summary

This paper investigates the structural properties of the complement of a finite union of convex sets in Euclidean space, focusing on three interrelated aspects: the number of isolated points, topological complexity (quantified via Betti numbers), and non-convexity. It addresses the minimal covering problem—covering the complement with a family of low-dimensional affine planes. We introduce a novel hierarchical plane-covering framework based on “local dimension matching,” yielding the first structural theorem for such covers. Integrating tools from algebraic topology, discrete geometry, and combinatorics, we derive tight upper bounds on the number of isolated points—in particular, an optimal bound in ℝ³—and significantly improve upon prior low-dimensional results. We also provide exact asymptotic estimates for the minimum cover size under various geometric configurations. The central innovation lies in establishing a rigorous correspondence between geometric singularities of the complement and its topological invariants, thereby shifting the analysis from existential constructions to intrinsic structural characterization.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Security and Privacy: Large-scale security measurements

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.

Problem

• Characterizing isolated points in complements of convex unions
• Covering complement sets with dimension-matching flats
• Establishing sharp bounds on covering sizes and isolated points

Innovation

• Studying isolated points in complements of convex unions
• Covering complements with flats matching local dimensions
• Obtaining sharp bounds on isolated points and covers