Duality of metric entropy

• 成果类型: Article
• 署名作者: Artstein, S; Milman, V; Szarek, SJ
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2004.159.1313
• 发表日期: 2004
• 页码: 1313-1328
• 关键词: numbers; Operators

摘要

For two convex bodies K and T in R-n, the covering number of K by T, denoted N(K, T), is defined as the minimal number of translates of T needed to cover K. Let us denote by K-o the polar body of K and by D the euclidean unit ball in R-n. We prove that the two functions of t, N(K, tD) and N(D, tK(o)), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K subset of R-m and over n is an element of N. In particular, this verifies the duality conjecture for entropy numbers of linear operators, posed by Pietsch in 1972, in the central case when either the domain or the range of the operator is a Hilbert space.