Maximum and minimum sets for some geometric mean values
Let \(v_ r\) denote the r-dimensional volume of the convex hull of \(r+1\) random points chosen independently and uniformly from a compact set K in \({\mathbb{R}}^ n\), and let h be a strictly increasing function. The author shows that for \(r=n\) (for \(r<n)\) the mean value of \(h\circ v_ r\) is minimal among all compact sets K of volume one, if K differs from an ellipsoid (from a ball) by a set of measure zero. This result is well- known only for convex K. Some other inequalities from geometric probability are extended to arbitrary compact sets. To prove these results, some integral inequalities for centrally symmetric quasiconvex and quasiconcave functions over compact rectangles are derived.
scientific article; zbMATH DE number 4178209