Papers › Maximal perimeter and maximal width of a convex small polygon
Maximal perimeter and maximal width of a convex small polygon
Christian Bingane
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A small polygon is a polygon of unit diameter. The maximal perimeter and the maximal width of a convex small polygon with n=2ˢ sides are unknown when s ≥4. In this paper, we propose an approach to construct convex small n-gons of large perimeter and large width when n=2ˢ with s≥2. Assuming the existence of an axis of symmetry, a convex small n-gon is described as a composition of n/2 and both its perimeter and its width are given as functions of a single variable. By selecting the composition that minimizes the violation of a cycle constraint by a particular solution, the n-gons constructed outperform the best n-gons found in the literature. For example, for n=64, the perimeter and the width obtained are within 10⁻²² and 10⁻¹² of the maximal perimeter and the maximal width, respectively. From our results, it appears that Mossinghoff's conjecture on the diameter graph of a convex small 2ˢ-gon with maximal perimeter is not true when s ≥4.
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