Conjecture on intersections of support cones implying homothetic ellipsoids
Establish that if L and K ⊂ ℝ^n (n ≥ 3) are convex bodies with L ⊂ int K and, for every boundary point x of K, there exist y ∈ bd K and a hyperplane Π such that S(L, x) ∩ S(L, y) = Π ∩ bd K, then L and K are homothetic ellipsoids. Here S(L, x) denotes the boundary of the cone C(L, x) generated by L with apex x.
References
This observation motives the following problem: \begin{conjecture}\label{tere} Let $L,K\subset R$ be convex bodies, $n\geq 3$. Suppose that $L\subset int K$ and, for every $x\in bd K$, there exists $y\in bd K$ and hyperplane $\Pi$ such that \begin{eqnarray}\label{nevada} S(L,x)\cap S(L,y)=\Pi \cap bd K. \end{eqnarray} Then $L$ and $K$ are homothetic ellipsoids. \end{conjecture}
— On characteristic properties of the ellipsoid in terms of circumscribed cones of a convex body
(2401.03983 - Morales-Amaya et al., 8 Jan 2024) in Conjecture (label tere), Section: Characterizations of the ellipsoid in terms of elliptical circumscribed cones of a convex body