The minimal volume of simplices containing a convex body

DANIEL GALICER; DAMIÁN PINASCO; MARIANO MERZBACHER

Buenos Aires

Congreso; Jornada de Análisis y Análisis Estocástico; 2017

The minimal volume of simplices containing a convex bodyDaniel GalicerAn old question in convex geometry is the following:How small is the the measure of Smin(K), the simplex of minimal volumeenclosing a given convex body K ⊂ Rn?Given a body K ⊂ Rn with barycenter at the origin, we show there is asimplex S inside K having also barycenter at the origin with ?large volume?.This is achieved using stochastic geometric techniques. Precisely, if K is inisotropic position, we present a method to find centered simplices with thatproperty that works with extremely high probability.As a consequence, we provide correct asymptotic estimates (when the di-mension n goes to infinity) on the aforementioned problem. Up to an absoluteconstant, the estimate cannot be lessened.Joint work with Dami ́an Pinasco and Mariano Merzbacher.