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**Constant mean curvature polytopes and hypersurfaces via projections.**
*(English)*
Zbl 1284.52015

Summary: Given a regular polytope in Euclidean space and an orthogonal projection to the complex plane, the function which assigns to each vertex its projected value satisfies a quadratic difference equation. The form of the equation is the same whatever the polytope, except for a real parameter \(\rho\) which varies from polytope to polytope. It is independent of the projection used and the size of the polytope. When we consider an orthogonal projection of a smooth hypersurface in Euclidean space, remarkably we find the same phenomena, namely that a smooth version of the equation is satisfied independently of the projection, where the parameter \(\rho\) depends only on the mean curvature. We therefore make an unconventional definition of a constant mean-curvature polytope as one which satisfies this same equation with \(\rho\) constant, independently of the orthogonal projection. We discuss some examples of constant mean curvature polytopes.

### MSC:

52B10 | Three-dimensional polytopes |

52B11 | \(n\)-dimensional polytopes |

39A14 | Partial difference equations |

53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |

### Keywords:

constant mean curvature hypersurface; constant mean curvature polytope; regular polytope; orthogonal projection
Full Text:
DOI

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