The minimal volume of simplices containing a convex body

PINASCO, DAMIÁN; MERZBACHER, MARIANO; GALICER, DANIEL

THE JOURNAL OF GEOMETRIC ANALYSIS

SPRINGER

Lugar: Berlin; Año: 2019 vol. 29 p. 717 - 732

1050-6926

Let K⊂ Rn be a convex body with barycenter at the origin. We show there is a simplex S⊂ K having also barycenter at the origin such that (vol(S)vol(K))1/n≥cn, 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⊂ Rn we show there is a simplex S enclosing Kwith the same barycenter such that(vol(S)vol(K))1/n≤dn,for some absolute constant d> 0. Up to the constant, the estimate cannot be lessened.