2017/07/11 by Daniel Galicer, Galicer, Daniel, Mariano Merzbacher +3 · 1 citation
Mathematics · #52A22 #52A23 #52A38 #52A40 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #math.FA #math.MG #msc:52A22 #msc:52A23 #msc:52A38 #msc:52A40
paper · pdf · doi:10.48550/arxiv.1707.03246
Some minor drafting errors were fixed
arxiv created 2019/07/17 · arxiv updated 2019/07/18
Let K ⊂ \mathbb 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 ≥ (c)/(√(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 Smin(K) of minimal volume enclosing a given convex body K ⊂ \mathbb Rn, fulfills the following inequality (\fracvol(Smin(K))vol(K))1/n ≤ d √(n), for some absolute constant d>0. Up to the constant, the estimate cannot be lessened.