Geometric realization of $\gamma$-vectors of 2-truncated cubes

TL;DR

This paper explores the geometric properties of 2-truncated cubes, establishing a unique correspondence between these polytopes and flag simplicial complexes that match their gamma-vectors, and confirming certain inequalities for these vectors.

Contribution

It introduces a unique function linking 2-truncated cubes to flag simplicial complexes with matching gamma-vectors, advancing understanding of their combinatorial structure.

Findings

Gamma-vectors of 2-truncated cubes satisfy Frankl-Furedi-Kalai inequalities.

A constructed function maps 2-truncated cubes to flag simplicial complexes with matching gamma-vectors.

The work extends previous results on the combinatorial properties of these polytopes.

Abstract

This paper continues investigation of the class of flag simple polytopes called 2-truncated cubes. It is an extended version of the short note Volodin (2012). A 2-truncated cube is a polytope obtained from a cube by sequence of truncations of codimension 2 faces. Constructed uniquely defined function which maps any 2-truncated cube to a flag simplicial complex with $f$-vector equal to $\gamma$-vector of the polytope. As a corollary we obtain that $\gamma$-vectors of 2-truncated cubes satisfy Frankl-Furedi-Kalai inequalities.