(beta)-distortion of some infinite graphs Academic Article uri icon

abstract

  • 2016 London Mathematical Society. A distortion lower bound of ((h){1/p}) is proved for embedding the complete countably branching hyperbolic tree of height h into a Banach space admitting an equivalent norm satisfying property () of Rolewicz with modulus of power type p-in (1) (in short property (). Also it is shown that a distortion lower bound of ( {1/p}) is incurred when embedding the parasol graph with levels into a Banach space with an equivalent norm with property ( p). The tightness of the lower bound for trees is shown adjusting a construction of Matouek to the case of infinite trees. It is also explained how our work unifies and extends a series of results about the stability under nonlinear quotients of the asymptotic structure of infinite-dimensional Banach spaces. Finally, two other applications regarding metric characterizations of asymptotic properties of Banach spaces, and the finite determinacy of bi-Lipschitz embeddability problems are discussed.
  • © 2016 London Mathematical Society. A distortion lower bound of Ω ((h){1/p}) is proved for embedding the complete countably branching hyperbolic tree of height h into a Banach space admitting an equivalent norm satisfying property (β) of Rolewicz with modulus of power type p-in (1) (in short property (β). Also it is shown that a distortion lower bound of Ω (ℓ {1/p}) is incurred when embedding the parasol graph with ℓ levels into a Banach space with an equivalent norm with property (β p). The tightness of the lower bound for trees is shown adjusting a construction of Matoušek to the case of infinite trees. It is also explained how our work unifies and extends a series of results about the stability under nonlinear quotients of the asymptotic structure of infinite-dimensional Banach spaces. Finally, two other applications regarding metric characterizations of asymptotic properties of Banach spaces, and the finite determinacy of bi-Lipschitz embeddability problems are discussed.

published proceedings

  • JOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES

altmetric score

  • 0.5

author list (cited authors)

  • Baudier, F. P., & Zhang, S.

citation count

  • 8

complete list of authors

  • Baudier, Florent P||Zhang, Sheng

publication date

  • January 2016

publisher