On Bezdek's conjecture for high-dimensional convex bodies with an aligned center of symmetry

Abstract

In 1999, K. Bezdek posed a conjecture stating that among all convex bodies in ${\mathbb{R}}^3$, ellipsoids and bodies of revolution are characterized by the fact that all their planar sections have an axis of reflection. We prove Bezdek's conjecture in arbitrary dimension $n\ge 3$, assuming only that sections passing through a fixed point have an axis of reflection, provided that the complementary invariant subspaces are all parallel to a fixed hyperplane. The result is proven in both orthogonal and affine settings.