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

Let $K$ be a convex body in an affine chart of the $n$ dimensional real Projective space ${\mathbb{R}\mathbb{P}}^n$, $n\ge 3$, let $H$ be a hyperplane which is not a support hyperplane of $K$ and let $p_1,p_2\in {\mathbb{R}\mathbb{P}}^n\setminus H$ 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.