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

Banach Center Publications (2005)

- Volume: 69, Issue: 1, page 81-90
- ISSN: 0137-6934

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topVladimir 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 -

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