## 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 $$0\ne M^n$$, 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.

### 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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