Metric Entropy of Convex Hulls
نویسنده
چکیده
Let T be a precompact subset of a Hilbert space. The metric entropy of the convex hull of T is estimated in terms of the metric entropy of T , when the latter is of order α = 2. The estimate is best possible. Thus, it answers a question left open in [LL] and [CKP]. 0.
منابع مشابه
Entropy of Absolute Convex Hulls in Hilbert Spaces
The metric entropy of absolute convex hulls of sets in Hilbert spaces is studied for the general case when the metric entropy of the sets is arbitrary. Under some regularity assumptions, the results are sharp.
متن کاملGelfand numbers and metric entropy of convex hulls in Hilbert spaces
We establish optimal estimates of Gelfand numbers or Gelfand widths of absolutely convex hulls cov(K) of precompact subsets K ⊂ H of a Hilbert space H by the metric entropy of the set K where the covering numbers N(K, ") of K by "-balls of H satisfy the Lorentz condition ∫ ∞ 0 ( log2N(K, ") )r/s d" <∞ for some fixed 0 < r, s ≤ ∞ with the usual modifications in the cases r = ∞, 0 < s < ∞ and 0 <...
متن کاملGeometric parameters in Learning Theory
3 Uniform measures of complexity 12 3.1 Metric entropy and the combinatorial dimension . . . . . . . . . 12 3.1.1 Binary valued classes . . . . . . . . . . . . . . . . . . . . . 13 3.1.2 Real valued classes . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Random averages and the combinatorial dimension . . . . . . . . 17 3.3 Phase transitions in GC classes . . . . . . . . . . . . . . . . . . ...
متن کاملMETRIC ENTROPY OF CONVEX HULLS IN TYPE p SPACES—THE CRITICAL CASE
Given a precompact subset A of a type p Banach space E, where p ∈ (1, 2], we prove that for every β ∈ [0, 1) and all n ∈ N sup k≤n k ′ (log k)ek(acoA) ≤ c sup k≤n k ′ (log k)ek(A) holds, where acoA is the absolutely convex hull of A and ek(.) denotes the kth dyadic entropy number. With this inequality we show in particular that for given A and β ∈ (−∞, 1) with en(A) ≤ n−1/p ′ (logn)−β for all n...
متن کاملOrthogonal metric space and convex contractions
In this paper, generalized convex contractions on orthogonal metric spaces are stablished in whath might be called their definitive versions. Also, we show that there are examples which show that our main theorems are genuine generalizations of Theorem 3.1 and 3.2 of [M.A. Miandaragh, M. Postolache and S. Rezapour, {it Approximate fixed points of generalized convex contractions}, Fixed Poi...
متن کامل