Empirical processes and random projections

In this note, we establish some bounds on the supremum of certain empirical processes indexed by sets of functions with the same L2 norm. We present several geometric applications of this result, the most important of which is a sharpening of the Johnson-Lindenstrauss embedding Lemma. Our results apply to a large class of random matrices, as we only require that the matrix entries have a subgaussian tail.