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.