In this case, g is called a \((\rho ,C)\)- coarse inverse of f. There are constants k, h(0) such that for all \(a,b,c,a',b',c'\in X\) we have (Bowditch ) A coarse median space is a triple, where ( X, d) is a metric space and is a ternary operator on X satisfying the following: 1.1 Bowditch’s definition of coarse median space Definition 1.1 We will provide the missing combinatorial framework by defining coarse median algebras. This prompts the question to what extent there could be a combinatorial characterisation of coarse medians mirroring the notion of a median algebra. In contrast, for a coarse median space the metric is an essential part of the data, as evidenced by the fact that almost any ternary algebra can be made into a coarse median space by equipping it with a bounded metric. The interaction between the geometry and combinatorics of a CAT(0) cube complex is mediated by the fact that the edge metric can be computed entirely in terms of the median. Centroid of a Triangle.png 1,465 × 1,044 93 KB. ApolloniusTheoremProof.svg 360 × 320 3 KB. Coarse median spaces as introduced by Bowditch provide a geometric coarsening of CAT(0) cube complexes which additionally includes \(\delta \)-hyperbolic spaces, mapping class groups and hierarchically hyperbolic groups. Media in category 'Median (geometry)' The following 42 files are in this category, out of 42 total. Its power stems from the beautiful interplay between the non-positively curved geometry of the space and the median algebra structure supported on the vertices as outlined by Roller. Gromov’s notion of a CAT(0) cube complex has played a significant role in major results in topology, geometry and group theory.
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