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Intersection depth and a Helly-type theorem for fractional transversals.

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Convex and discrete geometry; Algebraic topology; Combinatorics

video/mp4

Author affiliation: UNAM

2017-06-07

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61859

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/