Massachusetts Institute of Technology Lecturer : Michel X . Goemans

The aim of this lecture is to outline the gluing of embeddings at different scales described in James Lee’s paper Distance scales, embeddings, and metrics of negative type from SODA 2005 [1]. We begin by recalling some definitions. A map f : X → Y of metric spaces (X, dX ) and (Y, dY ) is said to be C-Lipschitz if dY (f(x), f(y)) ≤ CdX(x, y) for all x, y ∈ X. The infimum of all C such that f is C-Lipschitz is denoted by ‖f‖Lip. If f is bijective and ‖f‖Lip is finite, we say that f is bi-Lipschitz. The distortion of a bi-Lipschitz map is defined to be ‖f‖Lip‖f‖Lip. If f : X → Y is a 1-Lipschitz map such that for all x, y ∈ X satisfying τ ≤ dX(x, y) ≤ 2τ we have dY (f(x), f(y)) ≥ τ K ,