The Pentagonal Pizza Conjecture

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The pentagonal pizza conjecture says that it is impossible to cut a pentagonal pizza into eight slices by four concurrent straight lines that make four equal smaller angles alternating with four equal bigger angles, and such that all smaller angle slices have the same area and all bigger angle slices have the same area. The conjecture is proved to be true when the pizza is shaped as an ellipse, triangle, or quadrilateral, except for a square, and is proved not true for all n-gons, where . These results are closely related to the first theorem of convex geometry, Zindler’s theorem of 1920. Similar results are proved for perimeter instead of area.