Volume properties and some characterizations of ellipsoids in En+1

초록

Suppose that M is a strictly convex and closed hypersurface in En+1 with the origin o in its interior. We consider the homogeneous function g of positive degree d satisfying M = g(-1)(1). Then, for a positive number h the level hypersurface g(-1)(h) of g is a homothetic hypersurface of M with respect to the origin o. In this paper, for tangent hyperplanes Phi(h) to g(-1)(h) (0 < h < 1), we study the (n + 1)-dimensional volume of the region enclosed by Phi(h) and the hypersurface M, etc.. As a result, with the aid of the theorem of Blaschke and Deicke for proper affine hypersphere centered at the origin, we establish some characterizations for ellipsoids in En+1 . As a corollary, we extend Schneider's characterization for ellipsoids in E-3. Finally, for further study, we raise a question for elliptic paraboloids which was originally conjectured by Golomb.

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