A measure of axial symmetry of centrally symmetric convex bodies
Tom 121 / 2010
Colloquium Mathematicum 121 (2010), 295-306
MSC: Primary 52A10; Secondary 52A38.
DOI: 10.4064/cm121-2-12
Streszczenie
Denote by 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.
