On the mm-entropy of distributions of Gaussian processes

For a wide class of Banach spaces with Gaussian measure, it is shown that their Shannon entropy (mm-entropy) is closely related to the entropy of the corresponding kernel's ball and behaves in a certain range in the same way as the logarithm of the measure of small balls. The obtained results generalize the recent results of A. M. Vershik and M. A. Lifshits.