Ellipsoids and elliptic hyperboloids in the Euclidean space ${\mathbb{E}}^{n+1}$

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