Additive structures on $f$-vector sets of polytopes

TL;DR

This paper investigates the additive structures of $f$-vector sets of polytopes, revealing affine lattice spans, monoid embeddings, and approximate semigroup properties, especially in 4-polytopes, with implications for polytope construction.

Contribution

It introduces explicit descriptions of additive structures on $f$-vector sets, including monoid and approximate semigroup structures, and extends these concepts with geometric operations.

Findings

$f$-vector sets span affine lattices and form monoids in certain subclasses.

For 4-polytopes, the $f$-vector set forms an approximate affine semigroup.

Modified addition operations can be realized by polytope glueing, with potential semigroup structures in 4-polytopes.

Abstract

We show that the $f$-vector sets of $d$-polytopes have non-trivial additive structure: They span affine lattices and are embedded in monoids that we describe explicitly. Moreover, for many large subclasses, such as the simple polytopes, or the simplicial polytopes, there are monoid structures on the set of $f$-vectors by themselves: "addition of $f$-vectors minus the $f$-vector of the $d$-simplex" always yields a new $f$-vector. For general $4$-polytopes, we show that the modified addition operation does not always produce an $f$-vector, but that the result is always close to an $f$-vector. In this sense, the set of $f$-vectors of \emph{all} $4$-polytopes forms an "approximate affine semigroup." The proof relies on the fact for $d=4$ every $d$-polytope, or its dual, has a "small facet." This fails for $d>4$. We also describe a two further modified addition operations on $f$-vectors that can be geometrically realized by glueing corresponding polytopes. The second one of these may yield a semigroup structure on the $f$-vector set of all $4$-polytopes.