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Journal for Geometry and Graphics 16 (2012), No. 1, 041--046
Copyright Heldermann Verlag 2012

Characterizations of Euclidean Hyperspheres Under Relatively Normalized Convex Hypersurfaces

Georg Stamou
Dept. of Mathematics, Aristotle University, 54124 Thessaloniki, Greece

We treat convex hypersurfaces in the Euclidean space Rn+1 which are relatively normalized. The relative normalizations are either independent of geometric magnitudes of the considered convex hypersurface Φ or characterized by the fact that the corresponding support functions depend on elementary symmetric functions of the (Euclidean) principal curvatures of Φ. In the first case two characterizations of Euclidean hyperspheres are given via inequalities. In the second case it is proved that if the Pick-invariant vanishes identically, then Euclidean hyperspheres are obtained too.

Keywords: Convex hypersurfaces, relative normalizations, Pick-invariant, Euclidean hyperspheres.

MSC: 53A07

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