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Convex polytopes are the convex hulls of a finite set of points in Euclidean spaces. They have rich combinatorial, arithmetic, and metrical theory, and are related to toric varieties and to linear programming
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Cluster Variables for non-convex n-gons
Perhaps others will have suggestions, but I'm not aware of any cluster algebra papers that tackle this situation in the way you suggest.
One should be able to handle "diagonals without flips" as fr …