Duality of metric entropy

Shiri Artstein-Avidan, Vitali Milman, Stanisław Szarek
2004 Annals of Mathematics  
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 • 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 • ), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K ⊂ R n and over n ∈ N. In particular, this verifies the duality conjecture for entropy numbers of linear
more » ... ators, posed by Pietsch in 1972, in the central case when either the domain or the range of the operator is a Hilbert space.
doi:10.4007/annals.2004.159.1313 fatcat:grudiexdf5aotbmwhekmrnf3du