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arXiv · 1508.02380

Helly numbers of Algebraic Subsets of $\mathbb R^d$

J. A. De Loera·R. N. La Haye·D. Oliveros·E. Roldán-Pensado
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Abstract

We study $S$-convex sets, which are the geometric objects obtained as the intersection of the usual convex sets in $\mathbb R^d$ with a proper subset $S\subset \mathbb R^d$. We contribute new results about their $S$-Helly numbers. We extend prior work for $S=\mathbb R^d$, $\mathbb Z^d$, and $\mathbb Z^{d-k}\times\mathbb R^k$; we give sharp bounds on the $S$-Helly numbers in several new cases. We considered the situation for low-dimensional $S$ and for sets $S$ that have some algebraic structure, in particular when $S$ is an arbitrary subgroup of $\mathbb R^d$ or when $S$ is the difference between a lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz method we obtain colorful versions of many monochromatic Helly-type results, including several colorful versions of our own results.

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BibTeXRIS

J. A. De Loera, R. N. La Haye, D. Oliveros, E. Roldán-Pensado. 2015-08-10. Helly numbers of Algebraic Subsets of $\mathbb R^d$. https://arxiv.org/abs/1508.02380

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