Gaussian Samples Approach "Smooth Points" Slowest

Let K be the unit ball of the Hilbert space which generates a Gaussian measure μ on the real separable Banach space B. Some recent results establish that suitable normalized μ-Gaussian samples approach the smoothest points on the "boundary of K" slowest. As a partial explanation of this phenomenon we show that these points contain that portion of the boundary closest to points outside K. We also examine what happens if K is replaced by an arbitrary compact convex C in B, and attempt to characterize the set on the "boundary of C" which is closest to points outside C. Another result shows how this phenomenon characterizes B as a reflexive Banach space, and we also include some examples of interest.