  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.
