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.

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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
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preprint

A proof of Bezdek's conjecture on section symmetry

Aran S. Ziegler
Zenodo (CERN European Organization for Nuclear Research)
Point processes and geometric inequalities
preprint

A proof of Bezdek's conjecture on section symmetry

Aran S. Ziegler
preprint en

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.

Zenodo (CERN European Organization for Nuclear Research)
Auckland University of Technology (NZ)
Point processes and geometric inequalities
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