Ellipsoids and elliptic hyperboloids in the Euclidean space En+1En+1

We establish some characterizations of elliptic hyperboloids (resp., ellipsoids) in the (n+1)(n+1)-dimensional Euclidean space En+1En+1, using the n  -dimensional area of the sections cut-off by hyperplanes and the (n+1)(n+1)-dimensional volume of regions between parallel hyperplanes. We also give a few characterizations of elliptic paraboloids in the (n+1)(n+1)-dimensional Euclidean space En+1En+1.