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arXiv · 1506.02191

Positive-fraction intersection results and variations of weak epsilon-nets

Alexander Magazinov·Pablo Soberón
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Abstract

Given a finite set $X$ of points in $R^n$ and a family $F$ of sets generated by the pairs of points of $X$, we determine volumetric and structural conditions for the sets that allow us to guarantee the existence of a positive-fraction subfamily $F'$ of $F$ for which the sets have non-empty intersection. This allows us to show the existence of weak epsilon-nets for these families. We also prove a topological variation of the existence of weak epsilon-nets for convex sets.

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Alexander Magazinov, Pablo Soberón. 2015-06-06. Positive-fraction intersection results and variations of weak epsilon-nets. https://arxiv.org/abs/1506.02191

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