2023/09/22 by Goenka, Ritesh, Moore, Kenneth, Sun, Wen Rui +1
#52A10 #52A38 (Primary) 52A20 #52A41 (Secondary) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2309.12597
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 (2)/(41)(10 + 3 √(2)) ≈ 0.69476, thereby establishing a separation with the central symmetry measure. Moreover, we find a family of convex quadrilaterals with axiality approaching (1)/(3)(√(2)+1) ≈ 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.