Isometric and affine copies of a set in volumetric Helly results
Abstract
We show that for any compact convex set $K$ in $\mathbb{R}^d$ and any finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$, if the intersection of every sufficiently small subfamily of $\mathcal{F}$ contains an isometric copy of $K$ of volume $1$, then the intersection of the whole family contains an isometric copy of $K$ scaled by a factor of $(1-\varepsilon)$, where $\varepsilon$ is positive and fixed in advance. Unless $K$ is very similar to a disk, the shrinking factor is unavoidable. We prove similar results for affine copies of $K$. We show how our results imply the existence of randomized algorithms that approximate the largest copy of $K$ that fits inside a given polytope $P$ whose expected runtime is linear on the number of facets of $P$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2020
- DOI:
- 10.48550/arXiv.2010.04135
- arXiv:
- arXiv:2010.04135
- Bibcode:
- 2020arXiv201004135M
- Keywords:
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- Mathematics - Metric Geometry;
- Computer Science - Computational Geometry;
- Mathematics - Combinatorics
- E-Print:
- 10 pages, 2 figures