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

TL;DR

This paper demonstrates that associahedra minimize the $f$-vectors of secondary polytopes for planar point sets, extending previous results from triangulations to all regular subdivisions.

Contribution

It generalizes prior work by showing associahedra minimize $f$-vectors for all regular subdivisions, not just triangulations, in planar point sets.

Findings

Associahedra minimize $f$-vectors of secondary polytopes.

Regular triangulations suffice for the minimization result.

The minimization extends to all regular subdivisions.

Abstract

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: we show that regular triangulations are enough, and we extend the result to all regular subdivisions, graded by the dimension of their corresponding face in the secondary polytope.