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Frankl-Füredi-Kalai inequalities on the \(\gamma\)-vectors of flag nestohedra. (English) Zbl 1307.52006
A building set \(\mathcal{B}\) on \([n] = \{1,\ldots,n\}\) is a set of nonempty subsets of \([n]\) such that (i) for any non-disjoint \(I,J\in \mathcal{B}\) their union \(I\cup J\) is in \(\mathcal{B}\), and (ii) \(\mathcal{B}\) contains all singleton sets \(\{i\}\) for each \(i\in [n]\). The corresponding nestohedron \(P_{\mathcal{B}}\) is a polytope defined as the Minkowski sum \(P_{\mathcal{B}} := \sum_{I\in \mathcal{B}}\Delta_I\), where \(\Delta_I\) is the standard simplex with vertex set \(\{\tilde{e}_i : i\in I\}\) where \(\tilde{e}_i\) denotes the standard \(i\)-th basis vector for \({\mathbb{R}}^n\). A polytope is flag if every collection of pairwise intersecting facets has a nonempty intersection (i.e. the collection of the facets of \(P\) has the Helly property). A simplicial complex is flag if every set of pairwise adjacent vertices is a face.
The main result in this paper states that if \(P_{\mathcal{B}}\) is a flag nestohedron, then there is a corresponding flag simplicial complex \(\Gamma(\mathcal{B})\) whose \(f\)-vector \(f(\Gamma(\mathcal{B}))\) equals that of the \(\gamma\)-vector \(\gamma(P_{\mathcal{B}})\) of \(P_{\mathcal{B}}\). This shows, in particular, that \(\gamma(P_{\mathcal{B}})\) satisfies the Frankl-Füredi-Kalai inequalities [P. Frankl et al., Math. Scand. 63, No. 2, 169–178 (1988; Zbl 0651.05003)].

MSC:
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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