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Metric Entropy of Convex Hulls in Hilbert Spaces
Author(s) -
Carl Bernd
Publication year - 1997
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609397003044
Subject(s) - mathematics , convex hull , combinatorics , hull , regular polygon , hilbert space , convex set , metric space , radius , mathematical analysis , geometry , convex optimization , computer security , marine engineering , computer science , engineering
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 ρ n −σ . Then, for each n ∈ N, the convex hull of X can be covered by 2 n 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.