Duality of metric entropy

Abstract

AbstractFor two convex bodies Kand T in R n , the covering number of KbyT, denoted N(K,T), is defined as the minimal number of translates ofT needed to cover K. Let us denote by K ◦ the polar body of K andby D the euclidean unit ball in R n . We prove that the two functionsof 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. Inparticular, this verifies the duality conjecture for entropy numbers of linearoperators, posed by Pietsch in 1972, in the central case when either thedomain or the range of the operator is a Hilbert space. 1 Introduction For two convex bodies Kand T in R n , the covering number of Kby T,denoted N(K,T), is defined as the minimal number of translates of Tneeded to cover KN(K,T) = min{N: ∃x 1 ...x N ∈ R n , K⊂[ i≤N x i +T}.We denote by Dthe euclidean unit ball in R n . In this paper we prove thefollowing duality result for covering numbers.Theorem 1 (Main theorem) There exist two universal constants