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. 1991 Mathematics Subject Classification: 46B20, 46M35; 46B03, 54E35.