The classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences of closed convex hypersurfaces. This extended Minkowski problem is much more difficult since it essentially boils down to the question of solutions of certain Monge-Ampère equations of mixed type on the unit sphere of ℝn+1. In this paper, we mainly consider the uniqueness question and give first results.
We give a characterization of convex hypersurfaces with an equichordal point in terms of hedgehogs of constant width.
We characterize the linear space ℋ of differences of support functions of convex bodies of 𝔼² and we consider every h ∈ ℋ as the support function of a generalized hedgehog (a rectifiable closed curve having exactly one oriented support line in each direction). The mixed area (for plane convex bodies identified with their support functions) has a symmetric bilinear extension to ℋ which can be interpreted as a mixed area for generalized hedgehogs. We study generalized hedgehogs and we extend the...
Hedgehogs are a natural generalization of convex bodies of class C+ 2. After recalling some basic facts concerning this generalization, we use the notion of index to study differential and integral geometries of hedgehogs. As applications, we prove a particular case of the Tennis Ball Theorem and a property of normals to a plane convex body of constant width.
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