Abstract
We prove that the perimeter of any convex n-gons of diameter 1 is at most n2nsin (π/2n). Equality is attained here if and only if n has an odd factor. In the latter case, there are (up to congruence) only finitely many extremal n-gons. In fact, the convex n-gons of diameter 1 and perimeter n2n sin (π/2n) are in bijective correspondence with the solutions of a diophantine problem.
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Datta, B. A Discrete Isoperimetric Problem. Geometriae Dedicata 64, 55–68 (1997). https://doi.org/10.1023/A:1004997002327
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DOI: https://doi.org/10.1023/A:1004997002327