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. Thi...
Ausführliche Beschreibung

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 vol(S)vol(K)1/n≤dn,%$\begin{aligned} \left( \frac{\text {vol}(S)}{\text {vol}(K)}\right) ^{1/n} \le d \sqrt{n}, \end{aligned}%$for some absolute constant %$d>0%$. Up to the constant, the estimate cannot be lessened. Ausführliche Beschreibung