Approximation of a ball by random polytopes (Q1813616)

Let \(\omega_ d\) and \(\pi_ d\) denote the surface area and the volume of the \(d\)-dimensional unit ball \(B_ d\), and \(S_ n\) and \(V_ n\) the surface area and the volume of the convex hull of \(n\) random points chosen independently and uniformly from the boundary of \(B_ d\). The expected differences \(E(\omega_ d-S_ n)\) and \(E(\pi_ d-V_ n)\) both behave like \(c\cdot n^{-2/(d-1)}+\) lower order terms as \(n\to\infty\). The author gives explicit formulas for the coefficient \(c\) for both cases. The exponent of \(n\) in this asymptotic formula for random polytopes is the same as for best approximating polytopes. This contrast to the corresponding asymptotic formulas for convex polytopes generated by points randomly chosen from \(B_ d\), where the exponent is \(- 2/(d+1)\).