The minimal volume of simplices containing a convex body

Abstract

Let $K\subset {\mathbb{R}}^n$ be a convex body with barycenter at the origin. We show there is a simplex $S\subset K$ having also barycenter at the origin such that ${\left(\frac{vol(S)}{vol(K)}\right)}^{1/n}\ge \frac{c}{\sqrt{n}},$ where $c>0$ is an absolute constant. This is achieved using stochastic geometric techniques. Precisely, if $K$ is in isotropic position, we present a method to find centered simplices verifying the above bound that works with very high probability. As a consequence, we provide correct asymptotic estimates on an old problem in convex geometry. Namely, we show that the simplex $S_{min}(K)$ of minimal volume enclosing a given convex body $K\subset {\mathbb{R}}^n$, fulfills the following inequality $${\left(\frac{vol(S_{min}(K))}{vol(K)}\right)}^{1/n}\le d\sqrt{n},$$ for some absolute constant $d>0$. Up to the constant, the estimate cannot be lessened.

arXiv commentSome minor drafting errors were fixed