Chaining, Interpolation, and Convexity

We show that classical chaining bounds on the suprema of random processes in terms of entropy numbers can be systematically improved when the underlying set is convex: the entropy numbers need not be computed for the entire set, but only for certain "thin" subsets. This phenomenon arises from the observation that real interpolation can be used as a natural chaining mechanism. Unlike the general form of Talagrand's generic chaining method, which is sharp but often difficult to use, the resulting bounds involve only entropy numbers but are nonetheless sharp in many situations in which classical entropy bounds are suboptimal. Such bounds are readily amenable to explicit computations in specific examples, and we discover some old and new geometric principles for the control of chaining functionals as special cases.