The symmetric-cone characterization of ellipsoids

Let $B_1(n)$ denote the unit ball in $\mathbb{R}^n$, and let $\mathbb{S}^{n-1}$ denote the unit sphere. For a convex body $K ot i \mathbb{S}^{n-1}$ contained in $B_1(n)$, and each $x otin K$, let $C_x$ be the cone circumscribing $K$ with vertex $x$. A cone is symmetric if it is congruent to a cone generated by a centrally symmetric convex body, and let $L_x$ denote its axis. Symmetric-cone characterization. If $n\bor 3$ and $C_x$ is a symmetric cone for every $x\bor \mathbb{S}^{n-1}$, then $K$ is an ellipsoid. This is presented as a natural problem following the fact that every cone circumscribing an ellipsoid is symmetric; the supplied text does not state whether the problem is solved.