Collapsible polyhedra and median spaces

Vel, M. van de

doi:10.1090/s0002-9939-98-04413-x

Cited by 6 publications

(2 citation statements)

References 16 publications

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“…Such mixers are called symmetric. Symmetric mixers with additional algebraic properties are studied in the theory of median spaces (e.g., [25]).…”

Section: Definitions and Preliminary Resultsmentioning

confidence: 99%

Lipschitz means and mixers on metric spaces

Kovalev¹

2022

Preprint

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The standard arithmetic measures of center, the mean and median, have natural topological counterparts which have been widely used in continuum theory. In the context of metric spaces it is natural to consider the Lipschitz continuous versions of the mean and median. We show that they are related to familiar concepts of the geometry of metric spaces: the bounded turning property, the existence of quasisymmetric parameterization, and others.

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“…Such mixers are called symmetric. Symmetric mixers with additional algebraic properties are studied in the theory of median spaces (e.g., [25]).…”

Section: Definitions and Preliminary Resultsmentioning

confidence: 99%

Lipschitz means and mixers on metric spaces

Kovalev¹

2022

Preprint

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“…The proof of Theorem 1.8 involves a (non-straightforward) construction of a collapsible cubulation of the given collapsible polyhedron, which might be of interest in its own right. Another such construction (a more straightforward one) has been used to characterize collapsible polyhedra in the language of abstract convexity theory [48], and to establish the 'only if' part of Isbell's conjecture: a compact polyhedron is collapsible if and only if it is injectively metrizable [30], [49; Chapter VI]. (Isbell himself proved that the two conditions are equivalent for 2-polyhedra [21].…”

Section: A Embedding Contractible Polyhedra In Products Of Treesmentioning

confidence: 99%

Contractible polyhedra in products of trees and absolute retracts in products of dendrites

Melikhov

Zając

2013

Proc. Amer. Math. Soc.

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We show that a compact n-polyhedron PL embeds in a product of n trees if and only if it collapses onto an (n−1)-polyhedron. If the n-polyhedron is contractible and n = 3 (or n = 3 and the Andrews-Curtis Conjecture holds), the product of trees may be assumed to collapse onto the image of the embedding.In contrast, there exists a 2-dimensional compact absolute retract X such that X ×I k does not embed in any product of 2 + k dendrites for each k.

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Graphs of Some CAT(0) Complexes

Chepoi

2000

Advances in Applied Mathematics

214

251

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In this note, we characterize the graphs (1-skeletons) of some piecewise Euclidean simplicial and cubical complexes having nonpositive curvature in the sense of Gromov's CAT(0) inequality. Each such cell complex K is simply connected and obeys a certain flag condition. It turns out that if, in addition, all maximal cells are either regular Euclidean cubes or right Euclidean triangles glued in a special way, then the underlying graph G K is either a median graph or a hereditary modular graph without two forbidden induced subgraphs. We also characterize the simplicial complexes arising from bridged graphs, a class of graphs whose metric enjoys one of the basic properties of CAT (0) spaces. Additionally, we show that the graphs of all these complexes and some more general classes of graphs have geodesic combings and bicombings verifying the 1-or 2-fellow traveler property.

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Collapsible polyhedra and median spaces

Cited by 6 publications

References 16 publications

Lipschitz means and mixers on metric spaces

Lipschitz means and mixers on metric spaces

Contractible polyhedra in products of trees and absolute retracts in products of dendrites

Graphs of Some CAT(0) Complexes

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