## 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.)
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