  We show some characterizations of hyperspheres in the $(n+1)$-dimensional
Euclidean space ${\Bbb E}^{n+1}$ with intrinsic and extrinsic properties such
as the $n$-dimensional area of the sections cut off by hyperplanes, the
$(n+1)$-dimensional volume of regions between parallel hyperplanes, and the
$n$-dimensional surface area of regions between parallel hyperplanes. We also
establish two characterizations of elliptic paraboloids in the
$(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$ with the $n$-dimensional
area of the sections cut off by hyperplanes and the $(n+1)$-dimensional volume
of regions between parallel hyperplanes. For further study, we suggest a few
open problems.
