Geodesic
On $\mathrm{mm}$-entropy of a Banach space with a Gaussian measure
Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 3, pp. 532-543

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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.
Keywords: Gaussian measure, $\mathrm{mm}$-entropy, entropy of compact sets. 
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     author = {A. M. Vershik and M. A. Lifshits},
     title = {On $\mathrm{mm}$-entropy of a {Banach} space with a {Gaussian} measure},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {532--543},
     publisher = {mathdoc},
     volume = {68},
     number = {3},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2023_68_3_a5/}
}
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A. M. Vershik; M. A. Lifshits. On $\mathrm{mm}$-entropy of a Banach space with a Gaussian measure. Teoriâ veroâtnostej i ee primeneniâ, Tome 68 (2023) no. 3, pp. 532-543. http://geodesic.mathdoc.fr/item/TVP_2023_68_3_a5/