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

Vladimir Oliker. "A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature." Banach Center Publications 69.1 (2005): 81-90. <http://eudml.org/doc/282007>.

@article{VladimirOliker2005,
abstract = {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.},
author = {Vladimir Oliker},
journal = {Banach Center Publications},
keywords = {convex hypersurface; Gauss-Kronecker curavture; variational problem},
language = {eng},
number = {1},
pages = {81-90},
title = {A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature},
url = {http://eudml.org/doc/282007},
volume = {69},
year = {2005},
}

TY - JOUR
AU - Vladimir Oliker
TI - A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature
JO - Banach Center Publications
PY - 2005
VL - 69
IS - 1
SP - 81
EP - 90
AB - 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.
LA - eng
KW - convex hypersurface; Gauss-Kronecker curavture; variational problem
UR - http://eudml.org/doc/282007
ER -