Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 50, No. 2, pp. 541554 (2009) 

Minimumarea axially symmetric convex bodies containing a triangle and its measure of axial symmetryMarek Lassak and Monika NowickaInstitute of Mathematics, Polish Academy of Sciences, 'Sniadeckich 8, 00956 Warsaw, and University of Technology, Kaliskiego 7, 85796 Bydgoszcz, Poland, email: lassak@utp.edu.pl, University of Technology, Kaliskiego 7, 85796 Bydgoszcz, Poland, email: mnowicka@utp.edu.plAbstract: 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 $K^*_m$ is a measure of axial symmetry of $K$. A question is how to find a line $m$ in order to guarantee that $K^*_m$ be of the smallest possible area. A related task is to estimate ${\rm axs}(K)$ for the family of all convex bodies $K$. We give solutions for the classes of triangles, rightangled triangles and acute triangles. In particular, we prove that ${\rm axs}(T) > {1\over 2}\sqrt 2$ for every triangle $T$, and that this estimate cannot be improved in general. Keywords: triangle, convex body, axially symmetric body, mirror image, area, measure of axial symmetry Classification (MSC2000): 52A10, 52A38 Full text of the article:
Electronic version published on: 28 Aug 2009. This page was last modified: 28 Jan 2013.
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