Abstract.
We solve Talagrand’s entropy problem: the L 2-covering numbers of every uniformly bounded class of functions are exponential in its shattering dimension. This extends Dudley’s theorem on classes of {0,1}-valued functions, for which the shattering dimension is the Vapnik-Chervonenkis dimension. In convex geometry, the solution means that the entropy of a convex body K is controlled by the maximal dimension of a cube of a fixed side contained in the coordinate projections of K. This has a number of consequences, including the optimal Elton’s Theorem and estimates on the uniform central limit theorem in the real valued case.
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Oblatum 10-XII-2001 & 4-IX-2002¶Published online: 8 November 2002
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Mendelson, S., Vershynin, R. Entropy and the combinatorial dimension. Invent. math. 152, 37–55 (2003). https://doi.org/10.1007/s00222-002-0266-3
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DOI: https://doi.org/10.1007/s00222-002-0266-3