The cross-section body, plane sections of convex bodies and approximation of convex bodies. II
The authors continue their investigations begun in part I of this paper [ibid. 63, No. 3, 267-296 (1996; Zbl 0865.52005)]. They first study inequalities between volumes of \(k\)-dimensional projections of convex bodies and of projections of their k-dimensional sections, as well as dual versions of such results, involving sections of the bodies and sections of circumscribed cylinders. The next topic are Minkowski sums of random segments in a convex body \(K\subset\mathbb{R}^n\). Let \(x_1,y_1, \dots, x_n\), \(y_n(n\geq d)\) be independent uniform random points in \(K\) and let \(nD\) be the Minkowski sum of the segments with endpoints \(x_i,y_i(i=1,\dots,n)\). The authors study (in more general versions) the expected volume of \(D\), and for given volume of \(K\) they find sharp lower and, if \(d = 2\), also upper bounds. The results and methods are analogous to those in earlier investigations on convex hulls of random points. The final section presents some examples related to (volume) maximal cross sections of convex bodies.
scientific article; zbMATH DE number 1161898