On Axial Symmetry in Convex Bodies (2309.12597v1)

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.