$k$-dimensional transversals for fat convex sets
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916510344151040 |
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| author | Jung, Attila Pálvölgyi, Dömötör |
| author_facet | Jung, Attila Pálvölgyi, Dömötör |
| contents | We prove a fractional Helly theorem for $k$-flats intersecting fat convex sets. A family $\mathcal{F}$ of sets is said to be $ρ$-fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by $ρ$. We prove that for every dimension $d$ and positive reals $ρ$ and $α$ there exists a positive $β=β(d,ρ, α)$ such that if $\mathcal{F}$ is a finite family of $ρ$-fat convex sets in $\mathbb{R}^d$ and an $α$-fraction of the $(k+2)$-size subfamilies from $\mathcal{F}$ can be hit by a $k$-flat, then there is a $k$-flat that intersects at least a $β$-fraction of the sets of $\mathcal{F}$. We prove spherical and colorful variants of the above results and prove a $(p,k+2)$-theorem for $k$-flats intersecting balls. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_15646 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $k$-dimensional transversals for fat convex sets Jung, Attila Pálvölgyi, Dömötör Combinatorics Computational Geometry 52A35 We prove a fractional Helly theorem for $k$-flats intersecting fat convex sets. A family $\mathcal{F}$ of sets is said to be $ρ$-fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by $ρ$. We prove that for every dimension $d$ and positive reals $ρ$ and $α$ there exists a positive $β=β(d,ρ, α)$ such that if $\mathcal{F}$ is a finite family of $ρ$-fat convex sets in $\mathbb{R}^d$ and an $α$-fraction of the $(k+2)$-size subfamilies from $\mathcal{F}$ can be hit by a $k$-flat, then there is a $k$-flat that intersects at least a $β$-fraction of the sets of $\mathcal{F}$. We prove spherical and colorful variants of the above results and prove a $(p,k+2)$-theorem for $k$-flats intersecting balls. |
| title | $k$-dimensional transversals for fat convex sets |
| topic | Combinatorics Computational Geometry 52A35 |
| url | https://arxiv.org/abs/2311.15646 |