I do have a well-defined purpose in mind: to generate orthogonal projections of n-dimensional uniform polytopes into k-space, maximizing the distance between the image of any j-facet and that of any non-incident (k-1-j)-facet.  (Nearly all of the higher polytope images in Wikipedia are aligned for maximum symmetry instead.)  This means – or at least can be approximated by – excluding some bounded pieces of the Grassmannian, and finding the 'points' furthest from the excluded pieces.  A tractable mental image of the space that contains the exclusions would help!
 After another look at operator norm, I have a slightly better idea of the shape of my problem; just enough to see a flaw in the most obvious approach, and not enough to see a practical way around the flaw. —