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

TL;DR

This paper proves Bezdek's conjecture in all dimensions for convex bodies with sections passing through a fixed point having an axis of reflection, under specific symmetry conditions.

Contribution

It extends Bezdek's conjecture to higher dimensions under new symmetry assumptions, in both orthogonal and affine contexts.

Findings

Confirmed Bezdek's conjecture in arbitrary dimensions ≥ 3

Established symmetry characterization for convex bodies with reflective sections

Applicable in both orthogonal and affine geometric settings

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 $n\geq 3$, 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.