Bianchi–Gruber Conjecture: Symmetry of C(K,x) implies K is an ellipsoid

Prove that for a convex body K ⊂ ℝ^n (n ≥ 3) contained in the interior of the n-ball B(n), if for every x on the unit sphere S^{n−1} the cone C(K, x) generated by K with apex x is a symmetric cone, then K is an ellipsoid. Here C(K, x) = {x + λ(y − x) : y ∈ K, λ ≥ 0}.

Background

The paper seeks new characterizations of ellipsoids that relate to long-standing conjectures in convex geometry. One such conjecture by Bianchi and Gruber asks whether symmetry of the cones generated by a convex body from points on the unit sphere forces the body to be an ellipsoid.

The authors recall this conjecture explicitly and motivate their results as potential progress toward resolving it, focusing on properties of circumscribed cones and related geometric structures.