New notion of index for hedgehogs of R3R3 and applications

The aim of this paper is to introduce new tools for studying the following two important and difficult problems in R3R3: (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.