# zbMATH — the first resource for mathematics

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.)
Full Text:
##### References:
 [1] Aisbett, N., Inequalities between $$γ$$-polynomials of graph-associahedra, Electron. J. Comb., 19, 36, (2012) · Zbl 1243.05112 [2] Aisbett, N.: Gamma-vectors of edge subdivisions of the boundary of the cross polytope (2012). arXiv:1209.1789v1 [math.CO] [3] Buchstaber, V.M.; Volodin, V.D., Sharp upper and lower bounds for nestohedra, Izv. Math., 75, 1107-1133, (2011) · Zbl 1237.52006 [4] Erokhovets, N.Yu., Gal’s conjecture for nestohedra corresponding to complete bipartite graphs, Proc. Steklov Inst. Math., 266, 120-132, (2009) · Zbl 1227.52006 [5] Frohmader, A., Face vectors of flag complexes, Isr. J. Math., 164, 153-164, (2008) · Zbl 1246.05115 [6] Frankl, P.; Füredi, Z.; Kalai, G., Shadows of colored complexes, Math. Scand., 63, 169-178, (1988) · Zbl 0651.05003 [7] Gal, S.R., Real root conjecture fails for five and higher dimensional spheres, Discrete Comput. Geom., 34, 269-284, (2005) · Zbl 1085.52005 [8] Nevo, E.; Petersen, T.K., On $$γ$$-vectors satisfying the Kruskal-katona inequalities, Discrete Comput. Geom., 45, 503-521, (2010) · Zbl 1231.52009 [9] Nevo, E.; Petersen, T.K.; Tenner, B.E., The $$γ$$-vector of a barycentric subdivision, J. Comb. Theory, Ser. A, 118, 1364-1380, (2011) · Zbl 1231.05307 [10] Postnikov, A., Permutohedra, associahedra and beyond, Int. Math. Res. Not., 6, 1026-1106, (2009) · Zbl 1162.52007 [11] Postnikov, A.; Reiner, V.; Williams, L., Faces of generalized permutohedra, Doc. Math., 13, 207-273, (2008) · Zbl 1167.05005 [12] Volodin, V.D., Cubical realizations of flag nestohedra and a proof of gal’s conjecture for them, Usp. Mat. Nauk, 65, 188-190, (2010) · Zbl 1204.52014 [13] Volodin, V.D.: Geometric realization of the $$γ$$-vectors of 2-truncated cubes (2012). arXiv:1210.0398v1 [math.CO] · Zbl 1270.52013 [14] Volodin, V.D., Geometric realization of the $$γ$$-vectors of 2-truncated cubes, Russ. Math. Surv., 67, 582-584, (2012) · Zbl 1270.52013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.