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Kupavskii, Volostnov, and Yarovikov have recently shown that any set of $n$points in general position in the plane has at least as many (partial)triangulations as the convex $n$-gon. We generalize this in two directions: weshow that regular triangulations are enough, and we extend the result to allregular subdivisions, graded by the dimension of their corresponding face inthe secondary polytope

Associahedra minimize $f$-vectors of secondary polytopes of planar point sets