On $\mathrm{mm}$-entropy of a Banach space with a Gaussian measure

For a broad class of Banach spaces with Gaussian measure, we show that their entropy in the sense of Shannon (the $\mathrm{mm}$-entropy) is closely related to the entropy of the corresponding ellipsoid of concentration and behaves, in a certain range, as the logarithm of the measure of small balls. Relations between the $\mathrm{mm}$-entropy and the entropy of compact sets are also discussed in light of the classical works of Kolmogorov and Shannon.