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.

We show that the cyclohedron (Bott-Taubes polytope) W n arises as the polar dual of a Kantorovich-Rubinstein polytope KR(ρ), where ρ is an explicitly described quasimetric (asymmetric distance function) satisfying strict triangle inequality. From a broader perspective, this phenomenon illustrates the relationship between a nestohedron ∆ F (associated to a building set F) and its non-simple deformation ∆ F , where F is an irredundant or tight basis of F (Definition 21). Among the consequences are a new proof of a recent result of Gordon and Petrov (Arnold Math. J. 3 (2), 205-218 (2017)) about f -vectors of generic Kantorovich-Rubinstein polytopes and an extension of a theorem of Gelfand, Graev, and Postnikov, about triangulations of the type A, positive root polytopes.

Motivated by classical Euler's Tonnetz, we introduce and study the combinatorics and topology of more general simplicial complexes T onn n,k (L) of "Tonnetz type". Out main result is that for a sufficiently generic choice of parameters the generalized tonnetz T onn n,k (L) is a triangulation of a (k − 1)-dimensional torus T k−1 . In the proof we construct and use the properties of a discrete Abel-Jacobi map, which takes values in the torusk−1 is a permutohedral lattice.

A Bier sphere Bier(K) = K * ∆ K • , defined as the deleted join of a simplicial complex and its Alexander dual K • , is a purely combinatorial object (abstract simplicial complex). Here we study a hidden geometry of Bier spheres by describing their natural geometric realizations, compute their volume, describe an effective criterion for their polytopality, and associate to K a natural fan F an(K), related to the Braid fan. Along the way we establish a connection of Bier spheres of maximal volume with recent generalizations of the classical Van Kampen-Flores theorem and clarify the role of Bier spheres in the theory of generalized permutohedra.

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