Dichotomies, structure, and concentration in normed spaces Academic Article uri icon

abstract

  • © 2018 Elsevier Inc. We use probabilistic, topological and combinatorial methods to establish the following deviation inequality: For any normed space X=(Rn,‖⋅‖) there exists an invertible linear map T:Rn→Rn with P(|‖TG‖−E‖TG‖|>εE‖TG‖)≤Cexp⁡(−cmax⁡{ε2,ε}log⁡n),ε>0, where G is the standard n-dimensional Gaussian vector and C,c>0 are universal constants. It follows that for every ε∈(0,1) and for every normed space X=(Rn,‖⋅‖) there exists a k-dimensional subspace of X which is (1+ε)-Euclidean and k≥cεlog⁡n/log⁡[Formula presented]. This improves by a logarithmic on ε term the best previously known result due to G. Schechtman.

author list (cited authors)

  • Paouris, G., & Valettas, P.

citation count

  • 2

publication date

  • July 2018