On Axial Symmetry in Convex Bodies
For mathematicians studying convex geometry, this provides the first nontrivial lower bound for axiality that differs from the central symmetry measure, though the improvement is incremental.
The paper improves the lower bound for axial symmetry (axiality) of plane convex bodies from 2/3 to approximately 0.69476, establishing a separation from the Kovner-Besicovitch measure of central symmetry. It also provides a family of quadrilaterals with axiality approaching about 0.80474 and improves bounds for related measures.
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.