Diameter graphs of polygons and the proof of a conjecture of Graham |
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Authors: | Jim Foster |
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Institution: | Department of Mathematics, Weber State University, Ogden, UT 84408-1702, USA |
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Abstract: | We show that for an n-gon with unit diameter to have maximum area, its diameter graph must contain a cycle, and we derive an isodiametric theorem for such n-gons in terms of the length of the cycle. We then apply this theorem to prove Graham's 1975 conjecture that the diameter graph of a maximal 2m-gon (m?3) must be a cycle of length 2m−1 with one additional edge attached to it. |
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Keywords: | Discrete geometry Extremal polygons Diameter graph Isodiametric theorem Reuleaux polygons |
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