2022/03/01 by Catrina, Florin, Ostrovska, Sofiya, Ostrovskii, Mikhail I.
#30L05 #46B07 #46B85 #51F30 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2203.00166
The main result of the paper: Given any ε>0, every locally finite subset of ℓ2 admits a (1+ε)-bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which are of independent interest: (1) A direct sum of two finite-dimensional Euclidean spaces contains a sub-sum of a controlled dimension which is ε-close to a direct sum with respect to a 1-unconditional basis in a two-dimensional space. (2) For any finite-dimensional Banach space Y and its direct sum X with itself with respect to a 1-unconditional basis in a two-dimensional space, there exists a (1+ε)-bilipschitz embedding of Y into X which on a small ball coincides with the identity map onto the first summand and on a complement of a large ball coincides with the identity map onto the second summand.