Metric Entropy of Convex Hulls in Hilbert Spaces

Abstract

We show in this note the following statement which is an improvement over 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 constants ρ, σ>0, and for each n∈ N, the set X can be covered by at most n balls of radius ρn−σ. Then, for each n∈ N, the convex hull of X can be covered by 2n balls of radius c n − 1 2 − σ . The estimate is best possible for all n∈ N, apart from the value c=c(ρ, σ, X). In other words, let N(ε, X), ε>0, be the minimal number of balls of radius ε covering the set X. Then the above result is equivalent to saying that if N(ε, X)=O(ε−1/σ) as ε↓0, then for the convex hull conv (X) of X, N(ε, conv (X)) =O(exp(ε−2/(1 2σ))). Moreover, we give an interplay between several covering parameters based on coverings by balls (entropy numbers) and coverings by cylindrical sets (Kolmogorov numbers). 1991 Mathematics Subject Classification 41A46.