Displaying similar documents to “A De Bruijn-Erdős theorem for 1 - 2 metric spaces”

Certain contact metrics satisfying the Miao-Tam critical condition

Dhriti Sundar Patra, Amalendu Ghosh (2016)

Annales Polonici Mathematici

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We study certain contact metrics satisfying the Miao-Tam critical condition. First, we prove that a complete K-contact metric satisfying the Miao-Tam critical condition is isometric to the unit sphere S 2 n + 1 . Next, we study (κ,μ)-contact metrics satisfying the Miao-Tam critical condition.

The local metric dimension of a graph

Futaba Okamoto, Bryan Phinezy, Ping Zhang (2010)

Mathematica Bohemica

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For an ordered set W = { w 1 , w 2 , ... , w k } of k distinct vertices in a nontrivial connected graph G , the metric code of a vertex v of G with respect to W is the k -vector code ( v ) = ( d ( v , w 1 ) , d ( v , w 2 ) , ⋯ , d ( v , w k ) ) where d ( v , w i ) is the distance between v and w i for 1 ≤ i ≤ k . The set W is a local metric set of G if code ( u ) ≠ code ( v ) for every pair u , v of adjacent vertices of G . The minimum positive integer k for which G has a local metric k -set is the local metric dimension lmd ( G ) of G . A local metric set of G of cardinality lmd ( G ) is a local metric basis of G . We characterize all nontrivial...

Bilipschitz embeddings of metric spaces into euclidean spaces.

Stephen Semmes (1999)

Publicacions Matemàtiques

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When does a metric space admit a bilipschitz embedding into some finite-dimensional Euclidean space? There does not seem to be a simple answer to this question. Results of Assouad [A1], [A2], [A3] do provide a simple answer if one permits some small ("snowflake") deformations of the metric, but unfortunately these deformations immediately disrupt some basic aspects of geometry and analysis, like rectifiability, differentiability, and curves of finite length. Here we discuss a (somewhat...

Note on bi-Lipschitz embeddings into normed spaces

Jiří Matoušek (1992)

Commentationes Mathematicae Universitatis Carolinae

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Let ( X , d ) , ( Y , ρ ) be metric spaces and f : X → Y an injective mapping. We put ∥ f ∥ Lip = sup { ρ ( f ( x ) , f ( y ) ) / d ( x , y ) ; x , y ∈ X , x ≠ y } , and dist ( f ) = ∥ f ∥ Lip . ∥ f - 1 ∥ Lip (the of the mapping f ). We investigate the minimum dimension N such that every n -point metric space can be embedded into the space ℓ ∞ N with a prescribed distortion D . We obtain that this is possible for N ≥ C ( log n ) 2 n 3 / D , where C is a suitable absolute constant. This improves a result of Johnson, Lindenstrauss and Schechtman [JLS87] (with a simpler proof). Related results for embeddability into ℓ p N are obtained by a similar method. ...