Some characterizations of spheres and elliptic paraboloids II

TL;DR

This paper provides new characterizations of hyperspheres and elliptic paraboloids in higher-dimensional Euclidean spaces based on properties of sections and regions between hyperplanes, advancing geometric understanding.

Contribution

It introduces novel intrinsic and extrinsic characterizations of hyperspheres and elliptic paraboloids using hyperplane sections and volumes in higher dimensions.

Findings

Characterizations of hyperspheres via hyperplane sections and volumes.

Characterizations of elliptic paraboloids using similar geometric properties.

Proposes open problems for further research in geometric characterizations.

Abstract

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.