It is known that if finite subsets of a locally finite metric space $M$ admit bilipschitz embeddings into a Banach space $X$ with uniformly bounded distortions, say all distortions are $\le C$, then $M$ admits a bilipschitz embedding into $X$ with distortion $\le D\cdot C$, where $D$ is an absolute constant. One of the main goals of the talk is to show that for many Banach spaces, for example for such spaces as $\ell_p$ ($p\ne 2,\infty$) the constant $D$ is equal to $1^+$ in the following sense: The statement above does not hold for $D=1$, but holds for any $D=1+\ep$ with $\ep>0$.

This is a joint work with Sofiya Ostrovska. This work was supported by NSF DMS-1700176.