Duality of metric entropy in Euclidean space

Abstract

Let K be a convex body in a Euclidean space, K° its polar body and D the Euclidean unit ball. We prove that the covering numbers N( K, tD) and N( D, tK°) are comparable in the appropriate sense, uniformly over symmetric convex bodies K, over t>0 and over the dimension of the space. 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. To cite this article: S. Artstein et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).