C. E. Garza-Hume, Maricarmen C. Jorge, and Arturo Olvera,
Areas and shapes of planar irregular polygons,
Forum Geometricorum, 18 (2018) 17--36.

Abstract. We study analytically and numerically the possible shapes and areas of planar irregular polygons with prescribed side-lengths. We give an algorithm and a computer program to construct the cyclic configuration with its circumcircle and hence the maximum possible area. We study quadrilaterals with a self-intersection and prove that not all area minimizers are cyclic. We classify quadrilaterals into four classes related to the possibility of reversing orientation by deforming continuously. We study the possible shapes of polygons with prescribed side-lengths and prescribed area.

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