Convex bodies with many elliptic sections

TL;DR

This paper characterizes ellipsoids among convex bodies by the presence of multiple elliptic sections through each boundary point, providing new geometric criteria for identifying ellipsoids.

Contribution

It introduces novel characterizations of ellipsoids based on the number and configuration of elliptic sections through boundary points.

Findings

Two normal elliptic sections through every boundary point characterize an ellipsoid.

Four pairwise non-tangent elliptic sections through each boundary point also characterize an ellipsoid.

Abstract

{We show in this paper that two normal elliptic sections through every point of the boundary of a smooth convex body essentially characterize an ellipsoid and furthermore, that four different pairwise non-tangent elliptic sections through every point of the $C^2$-differentiable boundary of a convex body also essentially characterize an ellipsoid.