Axial symmetry in convex bodies

Ritesh Goenka, Kenneth Moore, Wen Rui Sun and Ethan Patrick White

Vol. 341 (2026), No. 2, 275–303

DOI: 10.2140/pjm.2026.341.275

Abstract

For a two-dimensional convex body, the Kovner–Besicovitch measure of symmetry is defined as the volume ratio of the largest centrally symmetric body contained inside the body to the original body. A classical result states that the Kovner–Besicovitch measure is at least $\frac{2}{3}$ for every convex body and equals $\frac{2}{3}$ for triangles. Lassak showed that an alternative measure of symmetry, i.e., symmetry about a line (axiality) has a value of at least $\frac{2}{3}$ for every convex body. However, the smallest known value of the axiality of a convex body is around $0.81584$, achieved by a convex quadrilateral. We show that every plane convex body has axiality at least $\frac{2}{41}(10+3\sqrt{2})\approx 0.69476$, thereby establishing a separation with the central symmetry measure. Moreover, we find a family of convex quadrilaterals with axiality approaching $\frac{1}{3}(\sqrt{2}+1)\approx 0.80474$. We also establish improved bounds for a “folding” measure of axial symmetry for plane convex bodies, and for a generalization of axiality to high-dimensional convex bodies.

Keywords

convex body, axial symmetry, axiality, folding symmetry

Mathematical Subject Classification

Primary: 52A10, 52A38

Secondary: 52A20, 52A41

Milestones

Received: 14 May 2024

Revised: 11 September 2025

Accepted: 21 November 2025

Published: 23 March 2026