Bourgain’s Discretization Theorem

Bourgain’s discretization theorem asserts that there exists a universal constant C ∈ (0,∞) with the following property. Let X,Y be Banach spaces with dimX = n. Fix D ∈ (1,∞) and set δ = e−nCn . Assume that N is a δ-net in the unit ball of X and that N admits a bi-Lipschitz embedding into Y with distortion at most D. Then the entire space X admits a bi-Lipschitz embedding into Y with distortion at most CD. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain’s theorem. We also obtain an improvement of Bourgain’s theorem in the important case when Y = Lp for some p ∈ [1,∞): in this case it suffices to take δ = C−1n−5/2 for the same conclusion to hold true. The case p = 1 of this improved discretization result has the following consequence. For arbitrarily large n ∈ N there exists a family Y of n-point subsets of {1, . . . , n} ⊆ R such that if we write |Y | = N then any L1 embedding of Y , equipped with the Earthmover metric (a.k.a. transportation cost metric or minimumum weight matching metric) incurs distortion at least a constant multiple of √ log logN ; the previously best known lower bound for this problem was a constant multiple of √ log log logN .