Hedgehogs and Zonoids

Abstract

Let R be a continuous real function on the unit sphere Sn of (n+1)-dimensional Euclidean space Rn+1. We prove that the maph:Sn→R,p↦∫Sn|〈p, q〉|R(q)dσ(q),where 〈·, ·〉 is the standard inner product and σ the spherical Lebesgue measure, is of class C2. It follows that the boundaries of zonoids (resp. generalized zonoids), whose generating measure have a continuous density with respect to σ, can be considered as hedgehogs (envelopes parametrized by their Gauss map) with a C2 support function. We deduce a local property for such zonoids. We give a formula for the curvature function of the hedgehog defined by h and we deduce a necessary and sufficient condition for h being the support function of a convex body of class C2+. We define projection hedgehogs (resp. mixed projection hedgehogs) and interpret their support functions in terms of n-dimensional volume (resp. mixed volume). Finally, we consider the extension of the classical Minkowski problem to hedgehogs.