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On the Hp-theorem for hypersurfaces. (English) Zbl 0678.53004
Consider a smooth oriented hypersurface $$M^ n$$ of Euclidean space $$R^{n+1}$$. If we choose an origin $$0\neq M^ n$$, we can define on $$M^ n$$ two smooth functions, the distance function $$r=| x|$$ and the support function $$p=-<x,n>$$, where x is the position vector and n is the unit normal of $$M^ n$$. The constancy of r implies, even locally, that $$M^ n$$ is part of a hypersphere centered at 0. The present article shows that: Let M be a closed, connected and oriented hypersurface in $$R^{n+1}$$, with mean curvature H; assume that $$Hp=1$$. Then M is a hypersphere.
Reviewer: T.Hasanis
##### MSC:
 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces 53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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