Denote by Kₘ the mirror image of a planar convex body K in a straight line m. It is easy to show that K*ₘ = conv(K ∪ Kₘ) is the smallest by inclusion convex body whose axis of symmetry is m and which contains K. The ratio 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 axs(K) > 1/2√2 for every centrally symmetric convex body and that this estimate cannot be improved in general. We also give a formula for axs(P) for every parallelogram P.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm121-2-12,
author = {Marek Lassak and Monika Nowicka},
title = {A measure of axial symmetry of centrally symmetric convex bodies},
journal = {Colloquium Mathematicae},
volume = {120},
year = {2010},
pages = {295-306},
zbl = {1217.52001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm121-2-12}
}
Marek Lassak; Monika Nowicka. A measure of axial symmetry of centrally symmetric convex bodies. Colloquium Mathematicae, Tome 120 (2010) pp. 295-306. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm121-2-12/