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A variational solution of the A. D. Aleksandrov problem of existence of a convex polytope with prescribed Gauss curvature. (English) Zbl 1089.53048

Opozda, Barbara (ed.) et al., PDEs, submanifolds and affine differential geometry. Proceedings of the conference and autumn school, Bȩdlewo, Poland, September 23–27, 2003. Warsaw: Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications 69, 81-90 (2005).
Let a complete convex hypersurface \(F\) in \(\mathbb R^{n+1}\) be given in the form \(z=u( x) \), \(x\in \mathbb R^{n}\) with the \(z\)-axis directed inside the convex body bounded by \(F\). If \(U\subseteq \mathbb R^{n}\) is a Borel set, then \(\nu_{F}( U) \) denotes the Gauss spherical image of the set of points \(\{ ( x,z) \text{ }|\)
For the entire collection see [Zbl 1077.53002].

MSC:

53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
49K20 Optimality conditions for problems involving partial differential equations
35J65 Nonlinear boundary value problems for linear elliptic equations
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