A proof of Bezdek's conjecture on section symmetry
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.
Authors
- Aran S. Ziegler (ORCID: https://orcid.org/0009-0004-8313-9696)
Institutions
- Auckland University of Technology (NZ)
Publication Details
- Journal
- Zenodo (CERN European Organization for Nuclear Research)
- Published
- 2026-10-13
- DOI
- https://doi.org/10.5281/zenodo.23316397
- Primary Topic
- Point processes and geometric inequalities
- Type
- preprint