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

2025/01/10 by Alfonseca, M. Angeles, Zawalski, B.
#52A10 #FOS: Mathematics #Metric Geometry (math.MG) #Primary 52A20 #Secondary 22E45

paper · doi:10.48550/arxiv.2501.06337

Abstract

In 1999, K. Bezdek posed a conjecture stating that among all convex bodies in \mathbb R³, 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≥ 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.

Related

• Reflections of convex bodies and their sections
• A solution to Bezdek's conjecture
• Convex bodies with sections with hyperplanes of symmetry
• Reflections of planar convex bodies
• Convex bodies with pairs of sections associated by reflections