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Lipschitzian mappings and total mean curvature of polyhedral surfaces. I. (English) Zbl 0563.52008

The author studies finite pointsets \(\{q_ o,...,q_ n\}\) and \(\{p_ o,...,p_ n\}\) in Euclidean space \(E^ m\) satisfying a Lipschitz condition \(| q_ i-q_ j| \leq c\cdot | p_ i-p_ j|\) for each pair i,j. He shows that the Minkowski functional M of the convex hulls satisfies \(M(conv\{q_ i\})\leq c\cdot M(conv\{p_ i\})\). For a smooth boundary of a convex body in \(E^ 3\) M is the total mean curvature which in the polyhedral case can be expressed as \(\sum_{i}\ell_ i(\pi -\alpha_ i)\) where \(\ell_ i\) is the length of the i-th edge and \(\alpha_ i\) is the corresponding dihedral angle along this edge. For this functional he shows a formula for smooth variations which in particular implies that the Connelly spheres [see R. Connelly, Inst. Haut. Etud. Sci., Publ. Math. 47, 333-338 (1977; Zbl 0375.53034)] must flex with constant total mean curvature.
Reviewer: W.Kühnel

MSC:

52A22 Random convex sets and integral geometry (aspects of convex geometry)
53C65 Integral geometry

Citations:

Zbl 0375.53034
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References:

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