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Abstract
We show some characterizations of hyperspheres in the (n+1)-dimensional Euclidean space En+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 En+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.
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