Title:  Metric cotype in Banach spaces.

Speaker: Ohad Giladi, NYU Courant  

Abstract: 

Among the most well known local properties of Banach

spaces are the notions of Rademacher type and cotype. The

original definition of the two notions depends heavily on the

linear structure of the space. More specifically, in the

definition we bound - either from above or from below - the

average norm of a random linear combination of vectors. However,

in 1976 Ribe proved that if two Banach spaces are uniformly

homeomorphic, they must have the same Rademacher type and

cotype. One would therefore be motivated to seek for a

definition which depends solely on the geometric structure of

the space, namely on distances between points.

 

Recently, Mendel and Naor gave a full solution to this problem,

first for the case of cotype and shortly later for type. In

their solution they introduce an auxiliary parameter, which can

be thought of as a correction term (in a sense that we will

explain). Bounding the parameter from below is rather

straightforward, but obtaining an upper bound is more involved,

and our work is aimed at improving the previously known bound.

We will review our work, which is based on averaging functions

with values in Banach spaces, and show applications to 

embedding of discrete spaces in Banach spaces. 

Joint work with Manor Mendel and Assaf Naor.