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A new characterization of the sphere in R 3

Thomas Hasanis (1980)

Annales Polonici Mathematici

Let M be a closed connected surface in R 3 with positive Gaussian curvature K and let K I I be the curvature of its second fundamental form. It is shown that M is a sphere if K I I = c H K r , for some constants c and r, where H is the mean curvature of M.

A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature

Vladimir Oliker (2005)

Banach Center Publications

In his book on convex polytopes [2] A. D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in n + 1 , n ≥ 2, with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem [2] and the Gauss curvature problem [20]. In this paper we give a simple variational proof of existence for the A. D. Aleksandrov problem...

Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes

Alexander I. Bobenko, Ivan Izmestiev (2008)

Annales de l’institut Fourier

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...

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