User yves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:04:41Z http://mathoverflow.net/feeds/user/22357 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/91981/monge-ampere-equations-concavity/92043#92043 Answer by yves for Monge Ampere equations (concavity) yves 2012-03-23T21:54:52Z 2012-03-23T21:54:52Z <p>The classical Minkowski problem is that of existence, uniqueness and regularity of closed convex hypersurfaces of the Euclidean linear space R^(n+1) whose Gaussian curvature (in the sense of Gauss’ definition) is prescribed as a function of the outer normal vector. For C2+ -hypersurfaces (C2-hypersurfaces with positive Gaussian curvature), this Minkowski problem is equivalent to the question of solutions of certain Monge–Ampère equations of elliptic type on the unit sphere S^n of R^(n+1). This classical Minkowski problem has a natural extension to hedgehogs, that is to Minkowski differences H = K − L of closed convex hypersurfaces K, L in R^(n+1), at least if we restrict ourselves to hypersurfaces whose support functions are of class C^2.</p> <p>For n = 2, the problem is already very difficult: if R in C(S^2;R) changes sign on S^2, the question of existence, uniqueness and regularity of a hedgehog of which R is the curvature function boils down to the study of a Monge-Ampère equation of mixed type, a class of equations for which there is no global result but only local ones by Lin and Zuily.</p> <p>First partial studies of this problem have been given in :</p> <ul> <li>Yves Martinez-Maure, Uniqueness results for the Minkowski problem extended to hedgehogs, Central European Journal of Mathematic 10, 2012, 440-450.</li> <li>Yves Martinez-Maure, New notion of index for hedgehogs of R^3 and applications, in:Rigidity and related topics in Geometry, European Journal of Combinatorics, 31 (2010), 1037-1049. </li> </ul>