Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

M. Angeles Alfonseca, B. Zawalski

TL;DR

This work extends Bezdek's conjecture to high dimensions by introducing an aligned center of symmetry: if every hyperplane section through a fixed point $p$ is invariant under reflection about a 1-dimensional subspace and the complementary invariant subspaces align with a fixed hyperplane, then the convex body $K\subset\mathbb{R}^n$ is either a body of (affine) revolution with axis through $p$ or an ellipsoid. It develops a general proof scheme that reduces the problem to many-body-type analyses on codimension-1 slices and leverages a robust affine-to-orthogonal transition under alignment, yielding a cascade of results (thm.14, thm.16, thm.02, thm.21) that cover affine and orthogonal settings across $n\ge 3$ (with explicit handling of $n=3$ vs $n\ge 4$). The techniques combine representation theory of the orthogonal group, invariant-subspace analysis, and geometric tomography, offering a unified framework that connects Brunn-type results, false-center phenomena, and Bezdek-type conjectures in higher dimensions. These findings advance the understanding of symmetry from hyperplane sections to global body structure, with potential broad impact on geometric tomography and the study of convex bodies through their section symmetries.

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.

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

TL;DR

This work extends Bezdek's conjecture to high dimensions by introducing an aligned center of symmetry: if every hyperplane section through a fixed point is invariant under reflection about a 1-dimensional subspace and the complementary invariant subspaces align with a fixed hyperplane, then the convex body is either a body of (affine) revolution with axis through or an ellipsoid. It develops a general proof scheme that reduces the problem to many-body-type analyses on codimension-1 slices and leverages a robust affine-to-orthogonal transition under alignment, yielding a cascade of results (thm.14, thm.16, thm.02, thm.21) that cover affine and orthogonal settings across (with explicit handling of vs ). The techniques combine representation theory of the orthogonal group, invariant-subspace analysis, and geometric tomography, offering a unified framework that connects Brunn-type results, false-center phenomena, and Bezdek-type conjectures in higher dimensions. These findings advance the understanding of symmetry from hyperplane sections to global body structure, with potential broad impact on geometric tomography and the study of convex bodies through their section symmetries.

Abstract

In 1999, K. Bezdek posed a conjecture stating that among all convex bodies in , 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 , 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.
Paper Structure (26 sections, 31 theorems, 43 equations, 3 figures)

This paper contains 26 sections, 31 theorems, 43 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definitions and basic concepts
  3. Representations of the orthogonal group
  4. Pseudo- and quasi-center of symmetry
  5. Historical background and known results
  6. Point reflection
  7. Affine involutions
  8. Rotations about a subspace
  9. An aligned center of symmetry
  10. A general proof scheme
  11. Statements of the new results
  12. Passing from the affine to the orthogonal variant
  13. Proofs of the new theorems
  14. Many-body problems
  15. Proof of \ref{['thm:14']}
  16. ...and 11 more sections

Key Result

Theorem 1

Let $K\subset\mathbb R^n$, $n\geq 3$, be a convex body and let $p\in\mathbb R^n$. If all hyperplane sections $K\cap H$ passing through $p$ are invariant under (affine) reflection through a $1$-dimensional subspace of $H$ passing through $p$, and the complementary invariant subspaces are all parallel

Figures (3)

  • Figure 4.3: Reducing the problem of a $(-1)$-aligned quasi-center to a many-body problem in codimension $1$
  • Figure 6.7: Notations used in the proof of \ref{['lem:08']}
  • Figure 7.3: The implication graph representing relations between different variants of \ref{['con:06']}

Theorems & Definitions (65)

  • Theorem 1: cf. \ref{['thm:14']}
  • Theorem 2: cf. \ref{['thm:18']}
  • Theorem 3: cf. \ref{['thm:16']}
  • Theorem 4: cf. \ref{['thm:02']}
  • Theorem 5: cf. \ref{['thm:22']}
  • Definition 2.2: kirillov2017introduction
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 55 more