Packing lines, planes, etc.: packings in Grassmannian spaces

Abstract

We address the question: How should $N$ $n$-dimensional subspaces of $m$-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of $N, n,m$ are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe $n$-dimensional subspaces of $m$-space as points on a sphere in dimension $\half(m-1) (m+2)$, which provides a (usually) lower-dimensional representation than the Plücker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multi-dimensional data via Asimov's grand tour method.