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.