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 ≥ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc69-0-4,
author = {Vladimir Oliker},
title = {A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature},
journal = {Banach Center Publications},
volume = {68},
year = {2005},
pages = {81-90},
zbl = {1089.53048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc69-0-4}
}
Vladimir Oliker. A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature. Banach Center Publications, Tome 68 (2005) pp. 81-90. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc69-0-4/