Lift zonoids, random convex hulls and the variability of random vectors
Zonoids are (centrally symmetric) convex bodies in \(\mathbb{R}^d\) which are limits of finite sums of line segments (zonotopes). For each measure \(F\) on \(\mathbb{R}^d\) with finite absolute first moment, the function \[ h(u)= \int_{ \mathbb{R}^d} \max\bigl \{0,\langle x,u\rangle \bigr\}dF(x), \quad u\in \mathbb{R}^d, \tag{*} \] is the support function of a zonoid \(Z\). Moreover, \(Z\) uniquely determines \(F\), provided \(F\) is an even measure concentrated on the unit sphere \(S^{d-1}\). These classical facts are used by the authors to introduce and study lift zonoids. For this purpose, an arbitrary measure \(F\) is lifted to a measure \(\widehat F\) on the hyperplane \(\{1\}\times \mathbb{R}^d \subset \mathbb{R}^{d+1}\) (according to an information from the authors, \(\alpha(F)\) in equation (3.1) has to be replaced by 1). The zonoid \(\widehat Z(F)\subset \mathbb{R}^{d+1}\) generated by \(\widehat F\) is then called the lift zonoid of \(F\). After proving a number of structural results for lift zonoids (linearity, injectivity, continuity, behaviour with respect to projections), some applications are discussed. For a probability distribution \(F\), the volume \(V_n\) of \(n\) independent random points \(X_1, \dots, X_n\) (distributed as \(F)\) is considered and bounds for the expectation \(EV_n\) are given in terms of the volume of \(\widehat Z(F)\). A main application concerns a new order between measures (probability distributions) on \(\mathbb{R}^d\), which is induced by the inclusion order of the corresponding lift zonoids. For this lift zonoid order, various results are proved and connections to random determinants are emphasized.
scientific article; zbMATH DE number 1239895