Distorting Banach Spaces

A Banach space X with a norm ||•|| is called D-distortable if there is an equivalent norm ||•|| on X so for each infinite dimensional subspace Y of X there are vectors x,y\in Y with ||x||=||y||=1 and ||x||/||y||>D. A space is called arbitrarily distortable if it is D-distortable for every D>1. A result of R.C. James from the 1960s shows that the Banach spaces \ell_1 and c0 are not distortable for any D>1. Shortly after this V. Milman showed that if a Banach space does not contain any \ell_p or c0 it must have a subspace that is D distortable for some D>1. In the 1990s it was shown explicitly by Odell and Schlumprecht that Tsirelson's famous space (not containing any \ell_p) was D distortable for each D<2. In the 1990s there were several outstanding results concerning distortion. The construction, by Schlumprecht, of the first known arbitrarily distortable Banach space and the very unexpected result of Odell and Schlumprecht that \ell_p is arbitrarily distortable for p\in (1,∞) were both extremely influentially to the general theory. It is still an open question as to whether there is a Banach space that is distortable but not arbitrarily distortable. In particular, it is not known if Tsirelson's space is arbitrarily distortable. Recently there has been some renewed attention to this and other problems related to distortion. On the website MathOverFlow, W.T. Gowers and P. Dodos suggested a set of problems that quantify distortion in a subtle combinatorial way. In this talk, we will explain the solution to some of these problems and how the problems relate to descriptive set theory and potentially some deep combinatorial principles. Some of the work we will mention is joint with Ryan Causey and Pavlos Motakis.

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