Pablo Soberon, Baruch College, City University of New York

Abstract

Helly’s theorem gives a characterization of families of convex sets in R^d with non-empty intersection.  Its quantitative versions aim to characterize families of convex sets with “large” intersection, which can be done by giving lower bounds on its volume or diameter.  During this talk we will discuss several quantitative Helly-type theorems where the size of the convex sets is witnessed by a low-complexity subset.  For instance, we prove Helly-type theorems which conclude that the intersection of a family of convex sets contains an ellipsoid of volume one, or that it contains a box of diameter one.  We use methods from classic convexity and from topological combinatorics to obtain our results.