  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.
