A measure of axial symmetry of centrally symmetric convex bodies
Volume 121 / 2010
Colloquium Mathematicum 121 (2010), 295-306
MSC: Primary 52A10; Secondary 52A38.
DOI: 10.4064/cm121-2-12
Abstract
Denote by $K_m$ the mirror image of a planar convex body $K$ in a straight line $m$. It is easy to show that $K^*_m = {\rm conv}(K\cup K_m)$ is the smallest by inclusion convex body whose axis of symmetry is $m$ and which contains $K$. The ratio ${\rm axs}(K)$ of the area of $K$ to the minimum area of $K^*_m$ over all straight lines $m$ is a measure of axial symmetry of $K$. We prove that ${\rm axs}(K) > {1\over 2}\sqrt 2$ for every centrally symmetric convex body and that this estimate cannot be improved in general. We also give a formula for ${\rm axs}(P)$ for every parallelogram $P$.
