Quermaßintegrals and asymptotic shape of random polytopes in an isotropic convex body (Q1951493)

Let \(K\) be an isotropic convex body in \(\mathbb R^n\). For every \(N>n\) consider the random polytope \(K_N :=\mathrm{conv}(\{\pm x_1,\dots, \pm x_N \})\), where \(x_1, \dots, x_N\) are independent random points, uniformly distributed in \(K\). It is proved that if \(n^2 \leq N \leq e^{\sqrt{n}}\), then the normalized quermassintegrals \(Q_k(K_N)\) of \(K_N\) satisfy the asymptotic formula \(Q_k(K_N ) \equiv L_K \sqrt{\log(N)}\) for all \(1 \leq k \leq n\). From this fact, precise quantitative estimates on the asymptotic behaviour of basic geometric parameters of \(K_N\) are obtained.