A measure of axial symmetry of centrally symmetric convex bodies

Marek Lassak, and Monika Nowicka. "A measure of axial symmetry of centrally symmetric convex bodies." Colloquium Mathematicae 121.2 (2010): 295-306. <http://eudml.org/doc/286249>.

@article{MarekLassak2010,
abstract = {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.},
author = {Marek Lassak, Monika Nowicka},
journal = {Colloquium Mathematicae},
keywords = {convex body; axially symmetric body; mirror image; area; measure of axial symmetry},
language = {eng},
number = {2},
pages = {295-306},
title = {A measure of axial symmetry of centrally symmetric convex bodies},
url = {http://eudml.org/doc/286249},
volume = {121},
year = {2010},
}

TY - JOUR
AU - Marek Lassak
AU - Monika Nowicka
TI - A measure of axial symmetry of centrally symmetric convex bodies
JO - Colloquium Mathematicae
PY - 2010
VL - 121
IS - 2
SP - 295
EP - 306
AB - 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.
LA - eng
KW - convex body; axially symmetric body; mirror image; area; measure of axial symmetry
UR - http://eudml.org/doc/286249
ER -