On random intersections of two convex bodies. Appendix to: "Isoperimetry of waists and local versus global asymptotic convex geometries" by R.Vershynin

TL;DR

This paper establishes a polynomial bound on the diameter of the intersection of two symmetric convex bodies after random rotation, extending previous results on their sections' boundedness.

Contribution

It provides a polynomial bound on the diameter of the intersection of two convex bodies under random rotation, improving upon earlier bounds.

Findings

Polynomial bound on intersection diameter established

Extension of previous bounded section results

Implications for convex geometric analysis

Abstract

In the paper "Isoperimetry of waists and local versus global asymptotic convex geometries", it was proved that the existence of nicely bounded sections of two symmetric convex bodies K and L implies that the intersection of randomly rotated K and L is nicely bounded. In this appendix, we achieve a polynomial bound on the diameter of that intersection (in the ratio of the dimensions of the sections).