AN EXTENSION OF SCHNEIDER'S CHARACTERIZATION THEOREM FOR ELLIPSOIDS

초록

Suppose that M is a strictly convex hypersurface in the (n + 1)-dimensional Euclidean space En+1 with the origin o in its convex side and with the outward unit normal N. For a fixed point p & ISIN; M and a positive constant t, we put (13t the hyperplane parallel to the tangent hyperplane (13 at p and passing through the point q = p- tN(p). We con-sider the region cut from M by the parallel hyperplane (13t, and denote by Ip(t) the (n + 1)-dimensional volume of the convex hull of the region and the origin o. Then Schneider's characterization theorem for ellip-soids states that among centrally symmetric, strictly convex and closed surfaces in the 3-dimensional Euclidean space E3, the ellipsoids are the only ones satisfying Ip(t) = & phi;(p)t, where & phi; is a function defined on M. Recently, the characterization theorem was extended to centrally sym-metric, strictly convex and closed hypersurfaces in En+1 satisfying for a constant & beta;, Ip(t) = & phi;(p)t & beta;. In this paper, we study the volume Ip(t) of a strictly convex and complete hypersurface in En+1 with the origin o in its convex side. As a result, first of all we extend the characterization theorem to strictly convex and closed (not necessarily centrally symmetric) hypersurfaces in En+1 satisfying Ip(t) = & phi;(p)t & beta;. After that we generalize the character-ization theorem to strictly convex and complete (not necessarily closed) hypersurfaces in En+1 satisfying Ip(t) = & phi;(p)t & beta;.

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