Minimum-Area Axially Symmetric Convex Bodies containing a Triangle and its Measure of Axial Symmetry
نویسندگان
چکیده
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 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 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 axs(T ) > 1 2 √ 2 for every triangle T , and that this estimate cannot be improved in general. MSC 2000: 52A10, 52A38
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