$k$-dimensional transversals for fat convex sets
| Autor: | Jung, Attila, Pálvölgyi, Dömötör |
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| Rok vydání: | 2023 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We prove a fractional Helly theorem for $k$-flats intersecting fat convex sets. A family $\mathcal{F}$ of sets is said to be $\rho$-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 $\rho$. We prove that for every dimension $d$ and positive reals $\rho$ and $\alpha$ there exists a positive $\beta=\beta(d,\rho, \alpha)$ such that if $\mathcal{F}$ is a finite family of $\rho$-fat convex sets in $\mathbb{R}^d$ and an $\alpha$-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 $\beta$-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. Comment: In the current version, the previous results concerning finite families of balls have been generalized to finite families of fat convex sets. The results from previous versions regarding infinite families are presented in arXiv:2412.04066 |
| Databáze: | arXiv |
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