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

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Oliker V.

Banach Center Publications

2005

tom 69

nr 1

81-90

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 $ℝ^{n+1}$, 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 [1,2] in which the hypersurface in question is a polyhedral convex graph over the entire ℝⁿ, has a prescribed asymptotic cone at infinity, and whose integral Gauss-Kronecker curvature has prescribed values at the vertices. The functional that we use is motivated by the functional arising in the dual problem in the Monge-Kantorovich optimal mass transfer theory considered by W. Gangbo [13] and L. Caffarelli [11]. The presented treatment of the Aleksandrov problem is self-contained and independent of the Monge-Kantorovich theory.

Ograniczanie wyników

1 Banach Center Publications

1 Oliker V.

1 2005

Wyniki wyszukiwania

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