(Tel Aviv University)
"Coverings and Metric Entropy: Duality"
Abstract:A well known 30 year old conjecture in the field of Asymptotic Geometric Analysis is the duality of entropy numbers conjecture, which concerns the quantifying, in terms of covering numbers, of the classical fact that if a linear operator between two Banach Spaces is compact then so is its dual. The most important case (which is the one appearing in applications to probability theory, ergodic theory, learning theory and other fields) is when one of the two spaces is a Hilbert space.
In 2003, jointly with V.D. Milman and S.J. Szarek, we proved the duality conjecture in this case. More recently, in a joint work together also with N. Tomczak-Jaegermann, we generalized this theorem to a wide class of spaces (which include for example all uniformly convex and all uniformly smooth spaces). This generalization invloved a new notion of 'convexified packing' which is of independent interest.
In the talk I will discuss these notions and results, and present, as much as time permits, the main ideas of the proofs.