The singular values of a matrix A can be viewed as describing the geometry of AB, where AB is the image of the euclidean ball under the linear transformation A. In particular, AB is an elipsoid, and the singular values of A describe the length of its major axes.
More generally, what do the singular values of a matrix say about the geometry of the image of other objects? How about the unit L1 ball? This will be some polytope: is there some natural way to describe this shape in terms of singular values, or other properties of matrix A?