On the extremal structure of least upper bound norms and their dual Original Research Article

Given finite dimensional real or complex Banach spaces, E and F, with norms image and image, we denote by Nμν the least upper bound norm induced on image. Some results are given on the extremal structures of image, the unit ball of Nμν, of its polar image, and of image, which is the polar of the unit ball of the least upper bound norm Nμ°ν°. The exposed faces, the extreme points, and a large family of other faces of image and image are presented. It turns out that image is a subset of image; the set of tangency points of the surfaces of image and image is completely determined and represented as the union of the exposed faces of image which are normal to rank-one mappings. We determine sharp bounds on the ranks of mappings in these exposed faces.