Complements of finite unions of convex sets
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914008646287360 |
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| author | Keller, Chaya Perles, Micha A. |
| author_facet | Keller, Chaya Perles, Micha A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_19413 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complements of finite unions of convex sets Keller, Chaya Perles, Micha A. Combinatorics Computational Geometry 52A37 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. |
| title | Complements of finite unions of convex sets |
| topic | Combinatorics Computational Geometry 52A37 |
| url | https://arxiv.org/abs/2508.19413 |