Distances between Banach spaces

The main object of the paper is to study the distance between Banach spaces introduced by Kadets. For Banach spaces X and Y , the Kadets distance is defined to be the infimum of the Hausdorff distance d ( B X , B Y ) between the respective closed unit balls over all isometric linear embeddings of X and Y into a common Banach space Z . This is compared with the Gromov-Hausdorff distance which is defined to be the infimum of d ( B X , B Y ) over all isometric embeddings into a common metric space Z . We prove continuity type results for the Kadets distance including a result that shows that this notion of distance has applications to the theory of complex interpolation.