A geometric proof of some inequalities involving mixed volumes.
A stability estimate for the Aleksandrow-Fnechel inequality, with an application to mean curvature.
Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes
We present a constructive proof of Alexandrov’s theorem on the existence of a convex polytope with a given metric on the boundary. The polytope is obtained by deforming certain generalized convex polytopes with the given boundary. We study the space of generalized convex polytopes and discover a connection with weighted Delaunay triangulations of polyhedral surfaces. The existence of the deformation follows from the non-degeneracy of the Hessian of the total scalar curvature of generalized convex...
Around the Rogers-Shepard inequality.