A Quantitative Helly-type Theorem: Containment in a Homothet

Abstract

We introduce a new variant of quantitative Helly-type theorems: the minimal \emph{"homothetic distance"} of the intersection of a family of convex sets to the intersection of a subfamily of a fixed size. As an application, we establish the following quantitative Helly-type result for the \emph{diameter}. If $K$ is the intersection of finitely many convex bodies in ${\mathbb{R}}^d$, then one can select $2d$ of these bodies whose intersection is of diameter at most $(2d{)}^3\mathrm{diam}(K)$. The best previously known estimate, due to Brazitikos, is $c{d}^{11/2}$. Moreover, we confirm that the multiplicative factor $c{d}^{1/2}$ conjectured by Bárány, Katchalski and Pach cannot be improved.

Some typos fixed