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.
Categories: math.DG