Construction of Shapes Arising from the Minkowski Problem Using a Level Set Approach

  • Authors:

  • Li-Tien Cheng

  • Affiliations:

  • Department of Mathematics, University of California San Diego, La Jolla, California 92093. lcheng@math.ucsd.edu

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2003

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Abstract

The Minkowski problem asks a fundamental question in differential geometry whose answer is not only important in that field but has real world applications as well. We endeavor to construct the shapes that arise from the Minkowski problem by forming a PDE that flows an initial implicitly defined hypersurface to an approximation of the shape under the level set framework. Tools and ideas found in the various applications of level set methods are gathered to generate this PDE. Numerically, its solution is determined by incorporating high order finite difference schemes over the uniform grid available in the framework. Finally, we use our approach in various test cases to generate various shapes arising from different given data in the Minkowski problem.