Average section functions for star-shaped sets

Note on Star-shaped Sets

1. The aim of this note is to prove that if 717" is a compact subset of En and for some m every w-dimensional hyperplane through a fixed point pQEn intersects M along a nonempty acyclic set, lúm^n — 1, then M is star-shaped with respect to p, i.e., if aQM then the segment pa is contained in M. This theorem is a generalization of a theorem of Aumann [l]. We gave recently a proof of Aumann's theo...

Invariant Valuations on Star-Shaped Sets

The Brunn Minkowski theory of convex bodies and mixed volumes has provided many tools for solving problems involving projections and valuations of compact convex sets in Euclidean space. Among the most beautiful results of twentieth century convexity is Hadwiger's characterization theorem for the elementary mixed volumes (Quermassintegrals); (see [3, 5, 9]). Hadwiger's characterization leads to...

Modelling and simulating star-shaped random sets

Lévy particles provide a flexible framework for modelling and simulating threedimensional star-shaped random sets. The radial function of a Lévy particle arises from a kernel smoothing of a Lévy basis, and is associated with an isotropic random field on the sphere. If the kernel is proportional to a von Mises–Fisher density, or uniform on a spherical cap, the correlation function of the associa...

Stochastic Orderings Induced by Star-Shaped Functions

Convex and star-shaped sets associated with stable distributions

It is known that each symmetric stable distribution in Rd is related to a norm on Rd that makes Rd embeddable in Lp([0, 1]). In case of a multivariate Cauchy distribution the unit ball in this norm corresponds is the polar set to a convex set in Rd called a zonoid. This work exploits most recent advances in convex geometry in order to come up with new probabilistic results for multivariate stab...