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

TL;DR

This paper characterizes elliptic hyperboloids, ellipsoids, and paraboloids in high-dimensional Euclidean space using geometric measures like hyperplane sections and volumes, providing new geometric insights.

Contribution

It introduces novel characterizations of these quadratic surfaces in ${ m I ext{-}dimensional}$ Euclidean space based on geometric measures.

Findings

Characterizations of elliptic hyperboloids and ellipsoids using hyperplane sections.

Characterizations of elliptic paraboloids in high-dimensional space.

Geometric criteria involving sections and volumes for identifying these surfaces.

Abstract

We establish some characterizations of elliptic hyperboloids (resp., ellipsoids) in the $(n+1)$-dimensional Euclidean space ${\Bbb E}^{n+1}$, 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 ${\Bbb E}^{n+1}$.