Inventiones mathematicae

, Volume 152, Issue 1, pp 37–55 | Cite as

Entropy and the combinatorial dimension

  • S. Mendelson
  • R. Vershynin


We solve Talagrand’s entropy problem: the L2-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.


Entropy Limit Theorem Central Limit Central Limit Theorem Convex Body 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • S. Mendelson
    • 1
  • R. Vershynin
    • 2
  1. 1.Research School of Information Sciences and Engineering, The Australian National University, Canberra, ACT 0200, Australia (e-mail:
  2. 2.Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada (e-mail:

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