Random Points in Isotropic Convex Sets

Summary

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 = L²|Y|²for 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⊂ ℝⁿ 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