A characterization of the ellipsoid in terms of pairs of sections associated by a harmonic homology

TL;DR

This paper characterizes ellipsoids in projective space by the existence of harmonic homologies relating pairs of sections associated with interior points and a hyperplane.

Contribution

It establishes a new geometric characterization of ellipsoids using harmonic homologies and hyperplane sections in projective space.

Findings

If such harmonic homologies exist for all $(n-2)$-planes in a hyperplane, then $K$ is an ellipsoid.

The characterization applies to convex bodies in affine charts of $ ext{RP}^n$, $n \\geq 3$.

The result links harmonic homologies with the geometric structure of ellipsoids.

Abstract

Let $K$ be a convex body in an affine chart of the $n$ dimensional real Projective space $\mathbb{RP}^n$, $n \geq 3$, let $H$ be a hyperplane which is not a support hyperplane of $K$ and let $p_1,p_2 \in \mathbb{RP}^n \setminus H$ be two distinct interior points of $K$. In this work we prove that if for every $(n-2)$-plane $l \subset H$, there exists a harmonic homology, with plane $G$ and center $\tau$, such that $l\subset G$, $\tau \in H$ and which maps the hypersection of $K$ defined by aff$\{p_1, l\}$ onto the hypersection of $K$ defined by aff$\{p_2, l\}$, then $K$ is an ellipsoid.