Geometric inequalities for Poisson processes of convex bodies and cylinders (Q580813)

There are various ways to measure the geometric behaviour and the denseness of a random point process Y in the space of d-dimensional convex bodies. For two of these measures, viz. the intersection point density of Y and the visible volume of Y, \textit{J. A. Wieacker} [Probab. Theory Relat. Fields 71, 405-433 (1986; Zbl 0562.60015)] proved some sharp inequalities of isoperimetric type. In the present paper, sharp inequalities in the reverse directions are derived. Moreover, they are extended to the space of k-cylinders. A k- cylinder is a d-dimensional closed convex set which can be written as the sum of a k-dimensional affine subspace and a compact convex set. In the appendix the author gives a hint that may be helpful to prove a conjecture of \textit{C. M. Petty} [Proc. Conf. Convexity combinat. Geometry Univ. Oklahoma 1971, 26-41 (1971; Zbl 0245.52007)] concerning the volume of the projection body of a convex body.