Lift zonoids, random convex hulls and the variability of random vectors

Abstract

For a $d$-variate measure a convex, compact set in $R^{d+1}$, its lift zonoid, is constructed. This yields an embedding of the class of $d$-variate measures having finite absolute first moments into the space of convex, compact sets in $R^{d+1}$. The embedding is continuous, positive homogeneous and additive and has useful applications to the analysis and comparison of random vectors. The lift zonoid is related to random convex sets and to the convex hull of a multivariate random sample. For an arbitrary sampling distribution, bounds are derived on the expected volume of the random convex hull. The set inclusion of lift zonoids defines an ordering of random vectors that reflects their variability. The ordering is investigated in detail and, as an application, inequalities for random determinants are given.