Talk information

Date: Sunday, November 30, 2025
Time: 10:10–11:00
Place: Schreiber 309
Speaker: Shakhar Smorodinsky (Ben-Gurion University)
Title: Extended VC-dimension and Radon and Tverberg-type Theorems for Unions of Convex Sets

Abstract:

Radon’s theorem asserts that any set $P$ of at least $d+2$ points in $\mathbb{R}^d$ can be partitioned into two parts whose convex hulls intersect.

In this talk, I will present our recent result that settles a long-standing open problem posed by Gil Kalai in the 1970s.

We obtain a near-tight estimate, for every dimension $d$ and integer $s \ge 1$, the smallest number $f(d,s)$ such that any set of $f(d,s)$ points in $\mathbb{R}^d$ admits a partition $P=A \cup B$ with the property that any union of $s$ convex sets containing $A$ must intersect any union of $s$ convex sets containing $B$. The classical Radon theorem corresponds to the case $s=1$ and is equivalent to the fact that $f(d,1)=d+2$ .

We also extend our framework to a Tverberg-type theorem for unions of convex sets. Our proofs are combinatorial and our bounds extend to so-called abstract convexity spaces with certain separability conditions.

Joint work with Noga Alon.