A proof of Bezdek's conjecture on section symmetry

Abstract

We prove Bezdek’s 1999 conjecture on Euclidean reflection symmetry of planar sections. A compact convex body with nonempty interior in three-dimensional Euclidean space, whose every affine plane section admits an orthogonal reflection, is an ellipsoid or a Euclidean body of revolution. This addresses the unrestricted three-dimensional problem without imposing alignment or global extendability of sectional mirrors. The argument obtains coherence from the section hypothesis itself: finite convex jets give exact slab reflections, and an off-axis nonquadric boundary marker forces global meridional reflections. No boundary regularity or alignment of sectional axes is assumed.