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.


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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