Duality of metric entropy in Euclidean space.
The subject of this paper is the duality conjecture for entropy numbers going back to \textit{A. Pietsch} [Theorie der Operatorenideale, Friedrich-Schiller-Universität Jena (1972; Zbl 0238.46067)]. For two convex bodies \(K\) and \(T\) in \({\mathbb R}^{n}\), the covering number of \(K\) by \(T\), denoted by \(N(K,T)\), is defined as the minimal number of translates of \(T\) needed to cover \(K\). The conjecture says that there exist two universal constants \(a,b\geq 1\) such that for any dimension \(n\) and for any two \(o\)-symmetric convex bodies \(K\) and \(T\) in \({\mathbb R}^{n}\) one has \(\log N(T^{\circ },aK^{\circ })\leq b\log N(K,T)\) where \(A^{\circ }\) denotes the polar body of \(A\). Here this conjecture is asserted to be valid in the important case where one of the two bodies is an ellipsoid. There is a sketch of a proof, but the details are announced to appear elsewhere.
scientific article; zbMATH DE number 2034210