Embedding \(\mathbb S^n\) into \(\mathbb R^{n+1}\) with given integral Gauss curvature and optimal mass transport on \(\mathbb S^n\)

DOI10.1016/j.aim.2007.01.005zbMath1233.49024arXivmath/0701398OpenAlexW2592146265MaRDI QIDQ2370610

Vladimir I. Oliker

Publication date: 29 June 2007

Published in: Advances in Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0701398

zbMATH Keywords

convexityGauss curvatureMonge-Kantorovich interpretationoptimal mass transportfinding a convex body with prescribed integral Gauss curvature

Mathematics Subject Classification ID

Lattices and convex bodies in (n) dimensions (aspects of discrete geometry) (52C07) Variational problems in a geometric measure-theoretic setting (49Q20) Methods involving semicontinuity and convergence; relaxation (49J45) Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) (53C42)

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Cites Work

• The Minkowski problem for polytopes
• Bounds for hyperspheres of prescribed Gaussian curvature
• An estimate of the deformation of a closed convex surface under a change in its curvature
• The geometry of optimal transportation
• Existence and uniqueness of monotone measure-preserving maps
• Monge’s transport problem on a Riemannian manifold
• Existence of optimal maps in the reflector-type problems
• Polar factorization and monotone rearrangement of vector‐valued functions
• On the $L_{p}$-Minkowski problem
• Optimal Sobolev norms and the Lp Minkowski problem
• Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle
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