We provide a new quantitative version of Helly’s theorem: there exists an absolute constant \(\alpha >1\) with the following property. If \(\{P_i: i\in I\}\) is a finite family of convex bodies in \({\mathbb {R}}^n\) with \({\mathrm{int}} (\bigcap _{i\in I}P_i )\ne \emptyset \), then there exist \(z\in {\mathbb {R}}^n\), \(s\leqslant \alpha n\) and \(i_1,\ldots i_s\in I\) such that #R##N##R##N#$$\begin{aligned} z+P_{i_1}\cap \cdots \cap P_{i_s}\subseteq cn^{3/2}\Big (z+\bigcap _{i\in I}P_i\Big ), \end{aligned}$$#R##N##R##N#where \(c>0\) is an absolute constant. This directly gives a version of the “quantitative” diameter theorem of Barany, Katchalski and Pach, with a polynomial dependence on the dimension. In the symmetric case the bound \(O(n^{3/2})\) can be improved to \(O(\sqrt{n})\).
['Silouanos Brazitikos']
Quantitative Helly-Type Theorem for the Diameter of Convex Sets