A Lower Bound Theorem for Centrally Symmetric Simplicial Polytopes

Abstract

Stanley proved that for any centrally symmetric simplicial d-polytope P with d ≥ 3, g 2 (P ) ≥ d 2 − d. We provide a characterization of centrally symmetric simplicial dpolytopes with d ≥ 4 that satisfy this inequality as equality. This gives a natural generalization of the classical Lower Bound Theorem for simplicial polytopes to the setting of centrally symmetric simplicial polytopes.

A Tale of Centrally Symmetric Polytopes and Spheres

“…That is, for a fixed r ≤ d/2, is there a characterization of cs simplicial d-polytopes with g r = d r − d r−1 ? In a recent preprint [41], Klee, Nevo, Novik, and Zheng provide such a characterization in the r = 2 case and a conjectural characterization in the r > 2 case. Both statements strongly parallel the equality cases of Theorems 7.1 and 7.2.…”

“…We mention that [41,Conjecture 8.6] provides a more detailed version of the above conjecture and we refer our readers to Ziegler's book [83,Section 8.1] for the definition of a polytopal complex. Conjecture 7.6, if true, would imply the following weaker statement that is also wide open at present.…”

“…One of the reasons Conjecture 8.4 appears to be hard in the general case is that the links of cs complexes are usually not centrally symmetric, and so starting with cs 2-spheres as the base case (and proceeding by induction via the cone and gluing lemmas) does not work. Conjecture 8.4 of [41] proposes a statement about rigidity of graphs of (non cs) simplicial 2-spheres that, if true, will provide an appropriate base case, and imply Conjecture 8.4. In any case, as a step in our quest for a cs analog of the g-conjecture, it would be extremely interesting to shed any light on Conjectures 8.2 and 8.4 as well as to attempt to strengthen the inequality of Conjecture 8.2 in the spirit of results from [54, Theorem 5.3(i)] and [56].…”

“…Surprisingly very little is known, and not for the lack of effort. There is an analog of the LBT for cs simplicial polytopes established by Stanley [74] as well as a characterization of minimizers [41], and while it is plausible that the same results continue to hold for all cs simplicial spheres, the proofs remain elusive.…”

Free Cells in Hyperspaces of Graphs

“…Graphs have been widely and deeply studied (see [3][4][5][6][7]) and have proved to be an excellent tool for representing and modeling different structures in several areas of discrete mathematics and computation (see [8,9]). As far as hyperspace is concerned, there exist some works relating both subjects.…”