Quermassintegrals of a random polytope in a convex body
In this interesting paper the authors extend results of Blaschke, Groemer, and Macbeath on the \(p\)th moment of the volume of a random polytope contained in a convex body. Their main result is the following: Let \(K\subset\mathbb{R}^{n}\) be a convex body with volume \(\left|K\right|=1\), and let \(C(x_{1},\dots ,x_{N})\) denote the convex hull of \(N\geq n+1\) points \(x_{1},\dots ,x_{N}\) independently and uniformly chosen in \(K\). Then, for every continuous and strictly increasing function \(f:\mathbb{R}^{+}\mathbb{\rightarrow R}^{+}\), and for \(0\leq i\leq n\), the mean value \[ E(K,N,f\circ W_{i})=\int_{K}\dots \int_{K}f\left[ W_{i}(C(x_{1},\dots ,x_{N}))\right] dx_{1}\dots dx_{N} \] is minimal among all convex bodies \(K\) of volume \(1\) when \(K\) is a ball (\(W_{i}\) being the \(i\)th quermassintegral of a compact convex set). Moreover, the ball is the unique minimizer when \(1\leq i\leq n\). The proof is based on Steiner symmetrization, and on a well-known characterization of the euclidean ball.
scientific article; zbMATH DE number 1966466