On the regularity of the solution of the <i>n</i>dimensional Minkowski problem

ShiuYuen Cheng Shing-Tung Yau

Convex and Discrete Geometry mathscidoc:1912.43480

Communications on pure and applied mathematics, 29, (5), 495-516, 1976.9
Given a closed strictly convex hypersurface M in the Euclidean space I?"+*, the Gauss map of M defines a homeomorphism between M and the unit sphere S". Therefore the Gauss-Kronecker curvature of M can be transplanted via the Gauss map to a function defined on S". If this function is denoted by K, then Minkowski observed that K must satisfy the equation where xi are the coordinate functions on S". Minkowski then asked the converse of the problem. Namely, given a positive function K defined on S" satisfying the above integral conditions, can we find a closed strictly convex hypersurface whose curvature function is given by K? Minkowski solved the problem in the category of polyhedrons. Then AD Alexandrov and others solved the problem in general. However, this last solution does not provide any information about the regularity of the (unique) convex hypersurface even if we assume K is smooth. In the two
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@inproceedings{shiuyuen1976on,
  title={On the regularity of the solution of the <i>n</i>dimensional Minkowski problem},
  author={ShiuYuen Cheng, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203522804071044},
  booktitle={Communications on pure and applied mathematics},
  volume={29},
  number={5},
  pages={495-516},
  year={1976},
}
ShiuYuen Cheng, and Shing-Tung Yau. On the regularity of the solution of the <i>n</i>dimensional Minkowski problem. 1976. Vol. 29. In Communications on pure and applied mathematics. pp.495-516. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203522804071044.
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