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 $(\frac{\text {vol}(S)}{\text {vol}(K)})^{1/n} \ge \frac{c}{\sqrt{n}},$ 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 extremely high probability. By duality, given a convex body $K \subset {\mathbb {R}}^n$ we show there is a simplex S enclosing Kwith the same barycenter such that

$$\begin{aligned} \left( \frac{\text {vol}(S)}{\text {vol}(K)}\right) ^{1/n} \le d \sqrt{n}, \end{aligned}$$

for some absolute constant $d>0$. Up to the constant, the estimate cannot be lessened.