Depth with respect to a family of convex sets

Abstract

We propose a notion of depth with respect to a finite family $\mathcal{F}$ of convex sets in $\mathbb{R}^d$ which we call $\text{dep}_\mathcal{F}$. We begin showing that $\text{dep}_\mathcal{F}$ satisfies some expected properties for a measure of depth and that this definition is closely related to the notion of depth proposed by J. Tukey. We show that some properties of Tukey depth extend to $\text{dep}_\mathcal{F}$ and we point out some key differences. We then focus on the following centerpoint-type question: what is the best depth $\alpha_{d,k}$ that we can guarantee under the hypothesis that the family $\mathcal{F}$ is $k$-intersecting? We show a key connection between this problem and a purely combinatorial problem on hitting sets. The relationship is useful in both directions. On the one hand, for values of $k$ close to $d$ the combinatorial interpretation gives a good bound for $k$. On the other hand, for low values of $k$ we can use the classic Rado's centerpoint theorem to get combinatorial results of independent interest. For intermediate values of $k$ we present a probabilistic framework to improve the bounds and illustrate its use in the case $k\approx d/2$. These results can be though of as an interpolation between Helly's theorem and Rado's centerpoint theorem. As an application of these results we find a Helly-type theorem for fractional hyperplane transversals. We also give an alternative and simpler proof for a transversal result of A. Holmsen.