METRIC ENTROPY OF CONVEX HULLS IN HILBERT-SPACES

We show in this note the following statement which is an improvement o ver a result of R. M. Dudley and which is also of independent interest . Let X be a set of a Hilbert space with the property that there are c onstants rho, sigma > 0, and for each n is an element of N, the set X can be covered by at most n balls of radius rho n(-sigma). Then, for e ach n epsilon N, the convex hull of X can be covered by 2(n) balls of radius cn(1/2 sigma). The estimate is best possible for all n is an el ement of N, apart from the value c = c(rho, sigma, X). In other words, let N(epsilon, X), epsilon > 0, be the minimal number of balls of rad ius epsilon covering the set X. Then the above result is equivalent to saying that if N(epsilon,X)= O(E-1/sigma) as E down arrow 0, then for the convex hull conv(X) of X, N(E,conv(X))= O(exp(epsilon(-2/(1+2 sig ma)))). Moreover, we give an interplay between several covering parame ters based on coverings by balls (entropy numbers) and coverings by cy lindrical sets (Kolmogorov numbers).