Random Points in Isotropic Convex Sets

Abstract

Let K be a symmetric convex body of volume 1 whose inertia tensor is isotropic, i.e., for some constant L we have ∫ K ( x , y ) 2 dx = L 2 | Y | 2 for all y . It is shown that if m is about n (log n ) 3 then with high probability, this tensor can be approximately realised by an average over m independent random points chosen in K . Our aim is to prove the following fact: PROPOSITION. Let K ⊂ ℝ n be a convex centrally symmetric body of volume 1, in isotropic position, i.e., Fix δ > 0 and choose m random points X l , … , X m ∈ K, where