Cyclohedron and Kantorovich–Rubinstein Polytopes

Jevtić, Filip D.; Jelić, Marija; Živaljević, Rade T.

doi:10.1007/s40598-018-0083-4

Cited by 3 publications

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“…The same authors also examined "generic metric spaces" (see Definition 5.4), computing their f-vectors (which, in this class, only depend on the number of elements in the space). Further study of fundamental polytopes of generic metrics appeared in [12,13] especially around a connection with duals of cyclohedra and Bier spheres. The special case of fundamental polytopes of full trees (Definition 2.8) fits into the framework of symmetric edge polytopes, which are themselves in the focus of active research, see e.g.…”

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confidence: 99%

Fundamental polytopes of metric trees via parallel connections of matroids

Delucchi

Hoessly

2020

European Journal of Combinatorics

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We tackle the problem of a combinatorial classification of finite metric spaces via their fundamental polytopes, as suggested by Vershik in 2010 [24].In this paper we consider a hyperplane arrangement associated to every split pseudometric and, for tree-like metrics, we study the combinatorics of its underlying matroid.• We give explicit formulas for the face numbers of fundamental polytopes and Lipschitz polytopes of all tree-like metrics,• We characterize the metric trees for which the fundamental polytope is simplicial.1991 Mathematics Subject Classification. 05-XX, 92Bxx, 54E35.

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confidence: 99%

Fundamental polytopes of metric trees via parallel connections of matroids

Delucchi

Hoessly

2020

European Journal of Combinatorics

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“…The problem then [54,Problem 1.1] is to relate the number of faces (and their incidences) of the fundamental polytope to the structure of the metric space. This line of research has been taken up in the literature from different points of view, see [14,25,31,32]. Face numbers of fundamental polytopes were computed for a class of "generic" metric spaces by Gordon and Petrov [25,Introduction], and in [14] for "treelike" metric spaces.…”

Section: Fundamental Polytopes Of Metric Spacesmentioning

confidence: 99%

“…There are many constructions associating a polytope to a graph. In this article we focus on three of them: symmetric edge polytopes [28,29,46,47,39], adjacency polytopes [10,8,7] as well as Kantorovich-Rubinstein polytopes and Lipschitz polytopes [54,14,25,31,32].…”

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confidence: 99%

Many Faces of Symmetric Edge Polytopes

D’Alì

Delucchi

Michałek

2022

Electron. J. Combin.

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Symmetric edge polytopes are a class of lattice polytopes constructed from finite simple graphs. In the present paper we highlight their connections to the Kuramoto synchronization model in physics — where they are called adjacency polytopes — and to Kantorovich-Rubinstein polytopes from finite metric space theory. Each of these connections motivates the study of symmetric edge polytopes of particular classes of graphs. We focus on such classes and apply algebraic combinatorial methods to investigate invariants of the associated symmetric edge polytopes.

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Polytopal Bier Spheres and Kantorovich–Rubinstein Polytopes of Weighted Cycles

Jevtić

Timotijević

Živaljević

2019

Discrete Comput Geom

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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.

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Cyclohedron and Kantorovich–Rubinstein Polytopes

Cited by 3 publications

References 25 publications

Fundamental polytopes of metric trees via parallel connections of matroids

Fundamental polytopes of metric trees via parallel connections of matroids

Many Faces of Symmetric Edge Polytopes

Polytopal Bier Spheres and Kantorovich–Rubinstein Polytopes of Weighted Cycles

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