Duality of metric entropy

Abstract

For two convex bodies $K$ and $T$ in $\mathbb{R}^n$, 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 $K^{\circ}$ the polar body of $K$ and by $D$ the euclidean unit ball in $\mathbb{R}^n$. We prove that the two functions of $t$, $N(K,tD)$ and $N(D, tK^{\circ})$, are equivalent in the appropriate sense, uniformly over symmetric convex bodies $K \subset \mathbb{R}^n$ and over $n \in \mathbb{N}$. 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.

Authors

Shiri Artstein

School of Mathematical Science
Tel Aviv University
69461 Tel Aviv
Israel

Vitali Milman

School of Mathematical Science
Tel Aviv University
69461 Tel Aviv
Israel

Stanisław J. Szarek

Equipe d’Analyse Fonctionnelle
Université Paris VI
75252 Paris
France and
Department of Mathematics
Case Western Reserve University
Cleveland, OH 44106
United States