Volume properties and some characterizations of ellipsoids in En+1

Suppose that M is a strictly convex and closed hypersurface in E 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 Φh to g−1(h) (0 < h < 1), we study the (n + 1) -dimensional volume of the region enclosed by Φh 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 E . As a corollary, we extend Schneider’s characterization for ellipsoids in E . Finally, for further study, we raise a question for elliptic paraboloids which was originally conjectured by Golomb.