Open Collections

Description

We introduce the notion of \textit{intersection depth} for a finite family of convex sets $\mathcal{F}$ in $\mathbb{R}^d$. Specifically, we say that a point $p$ has \textit{intersection depth $m$ with respect to $\mathcal{F}$} if every hyperplane that contains $p$ intersects at least $m$ sets of $\mathcal{F}$. We study some nice properties of intersection depth and we relate it to other notions of depth in the literature. By imposing additional intersection hypothesis to the family $\mathcal{F}$, we show how to prove sharp centerpoint theorems for intersection depth. These results can be thought of as a refinement that interpolates between the classical Rado's centerpoint theorem and Helly's theorem. Finally, we use this result to get a new Helly-type theorem for fractional transversal hyperplanes that cannot be obtained from the well-studied $T(k)$ hypothesis.