Volume properties and some characterizations of ellipsoids in E n+1

Suppose that M is a strictly convex and closed hypersurface in En+1 with the origin o in its interior. Weconsider the homogeneous function g of positive degree d satisfying M = g−1(1). Then, for a positive number h thelevel hypersurface g−1(h) of g is a homothetic hypersurface of M with respect to the origin o. In this paper, for tangenthyperplanes Φh to g−1(h) (0 < h < 1), we study the (n + 1)-dimensional volume of the region enclosed by Φh andthe hypersurface M , etc.. As a result, with the aid of the theorem of Blaschke and Deicke for proper affine hyperspherecentered at the origin, we establish some characterizations for ellipsoids in En+1 . As a corollary, we extend Schneider’scharacterization for ellipsoids in E3. Finally, for further study, we raise a question for elliptic paraboloids which wasoriginally conjectured by Golomb.

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