Convex geometry of max-stable distributions

It is shown that max-stable random vectors in [0, infinity)(d) with unit Frechet marginals are in one to one correspondence with convex sets K in [0, 8)(d) called max-zonoids. The max-zonoids can be characterised as sets obtained as limits of Minkowski sums of cross-polytopes or, alternatively, as the selection expectation of a random cross-polytope whose distribution is controlled by the spectral measure of the max-stable random vector. Furthermore, the cumulative distribution function P{xi <= x} of a max-stable random vector xi with unit Frechet marginals is determined by the norm of the inverse to x, where all possible norms are given by the support functions of (normalised) max-zonoids. As an application, geometrical interpretations of a number of well-known concepts from the theory of multivariate extreme values and copulas are provided.