On Axial Symmetry in Convex Bodies

TL;DR

This paper investigates measures of axial symmetry in convex bodies, establishing new lower bounds for axiality in two-dimensional and high-dimensional cases, and exploring the relationship with central symmetry measures.

Contribution

It provides the first explicit lower bound for axiality of convex bodies in the plane and extends symmetry bounds to higher dimensions, improving upon previous results.

Findings

Every plane convex body has axiality at least approximately 0.69476.

Constructed convex quadrilaterals with axiality approaching approximately 0.80474.

Established improved bounds for folding measures and high-dimensional axial symmetry.

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 $2/3$ for every convex body and equals $2/3$ for triangles. Lassak showed that an alternative measure of symmetry, i.e., symmetry about a line (axiality) has a value of at least $2/3$ 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. Finally, we establish improved bounds for a generalization of axiality to high-dimensional convex bodies.