The minimal volume of simplices containing a convex body

TL;DR

This paper proves the existence of a large-volume simplex within any convex body in n-dimensional space, using stochastic methods, and provides asymptotic bounds on the minimal volume enclosing simplex.

Contribution

It introduces a stochastic geometric approach to find large simplices inside convex bodies and establishes asymptotic bounds on minimal enclosing simplices in high dimensions.

Findings

Existence of a centered simplex with volume ratio at least c/√n.

High probability method for finding such simplices in isotropic position.

Asymptotic estimate for minimal volume enclosing simplex proportional to √n.

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 $\left(\frac{vol(S)}{vol(K)}\right)^{1/n} \geq \frac{c}{\sqrt{n}},$ 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 $$\left(\frac{vol(S_{min}(K))}{vol(K)}\right)^{1/n} \leq d \sqrt{n},$$ for some absolute constant $d>0$. Up to the constant, the estimate cannot be lessened.