Polytopal Bier Spheres and Kantorovich–Rubinstein Polytopes of Weighted Cycles

Abstract

The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the 'Simplicial Steinitz problem'. It is known by an indirect and non-constructive argument that a vast majority of Bier spheres are non-polytopal. Contrary to that, we demonstrate that the Bier spheres associated to threshold simplicial complexes are all polytopal. Moreover, we show that all Bier spheres are starshaped. We also establish a connection between Bier spheres and Kantorovich-Rubinstein polytopes by showing that the boundary sphere of the KR-polytope associated to a polygonal linkage (weighted cycle) is isomorphic to the Bier sphere of the associated simplicial complex of "short sets".[MPV] (with J. Melleray and F. Petrov) and to Kantorovich himself, see [K-R] where the Kantorovich-Rubinstein norm µ − ν KR is introduced.Bier spheres Bier(K), where K 2 [n] is an abstract simplicial complex, are combinatorially defined triangulations of the (n − 2)-dimensional sphere S n−2 with interesting combinatorial and topological properties, [Lo1, M03]. These spheres are known to be shellable [BPSZ, ČD]. Moreover it is known, by an indirect and non-effective counting argument, that the majority of these spheres are non-polytopal, in the sense that they do not admit a convex polytope realization, see [M03, Section 5.6]. They also provide one of the most elegant proofs of the Van Kampen-Flores theorem [M03] and serve as one of the main examples of "Alexander complexes" [JNPZ].Threshold complexes are ubiquitous in mathematics and arise, often in disguise and under different names, in areas as different as cooperative game theory (quota complexes and simple games) and algebraic topology of configuration spaces (polygonal linkages, complexes of short sets) [CoDe, Far, GaPa].The following theorem establishes a connection between the boundary ∂KR(d L ) of the Kantorovich-Rubinstein polytope of a weighted cycle, and the Bier sphere of the threshold complex of "short sets" of the associated polygonal linkage.

Fundamental polytopes of metric trees via parallel connections of matroids

“…Since σ does not split any block of π ′ \ {A ⊔ B}, we have W ⊆ (v σ ) ⊥ . Therefore, the right-hand side of Equation (13)…”

Bier spheres of extremal volume and generalized permutohedra

“…The following proposition shows that the fan F an(K) is isomorphic to the radial fan associated to the starshaped realization R ±δ (Bier(K)) of the Bier sphere Bier(K), constructed in [7], Theorem 3.5. The reader is referred to Section 2 (see also [7]) for all undefined concepts and related facts. In particular the δ-realization is a special case of the b-realization from Section 2 where the vertices δ = {δ 1 , .…”

“…It turns out that the radial fan F an(K) of the starshaped embedding of the Bier sphere Bier(K), described in the proof of this result, is a coarsening of the Braid arrangement fan. This fact was not emphasized in [7], however it is interesting in itself and certainly deserves further study.…”

“…One of the main new results of [7] was the observation ([7, Theorem 3.1]) that each Bier sphere Bier(K), defined as a canonical triangulation of a (n − 2) sphere S n−2 associated to an abstract simplicial complex K 2 [n] , admits a starshaped embedding in R n−1 .…”

“…F an(K) = BierF an(K), the canonical or the Bier fan of K, is a complete, simplicial fan in H 0 ∼ = R n−1 , associated to a simplicial complex K 2 [n] . R ±δ (Bier(K)) is the canonical starshaped realization of Bier(K) described in [7,Theorem 3.1].…”