Minimum-area axially symmetric convex bodies containing a triangle and its measure of axial symmetry

Marek Lassak and Monika Nowicka

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 $K^*_m$ 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, right-angled 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

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