New notion of index for hedgehogs of and applications

The aim of this paper is to introduce new tools for studying the following two important and difficult problems in R 3 : (1) The Minkowski problem (to prescribe the Gauss curvature) for hedgehogs (i.e., for Minkowski differences of convex bodies); (2) The search for Sturm–Hurwitz type theorems (relating number of zeros to expansions in spherical harmonics). First, (1) we give a brief survey of hedgehog theory and a short introduction to these problems; (2) we recall briefly the main results already obtained (one of which is a counter-example to a conjecture of A.D. Alexandrov) and we explain why new tools are necessary for going further. Finally, we introduce a new notion of index for studying hedgehogs and we give first geometrical applications.