  This paper concerns the minimax center of a collection of linear subspaces.
When the subspaces are $k$-dimensional subspaces of $\mathbb{R}^n$, this can be
cast as finding the center of a minimum enclosing ball on a Grassmann manifold,
Gr$(k,n)$. For subspaces of different dimension, the setting becomes a disjoint
union of Grassmannians rather than a single manifold, and the problem is no
longer well-defined. However, natural geometric maps exist between these
manifolds with a well-defined notion of distance for the images of the
subspaces under the mappings. Solving the initial problem in this context leads
to a candidate minimax center on each of the constituent manifolds, but does
not inherently provide intuition about which candidate is the best
representation of the data. Additionally, the solutions of different rank are
generally not nested so a deflationary approach will not suffice, and the
problem must be solved independently on each manifold. We propose and solve an
optimization problem parametrized by the rank of the minimax center. The
solution is computed using a subgradient algorithm on the dual. By scaling the
objective and penalizing the information lost by the rank-$k$ minimax center,
we jointly recover an optimal dimension, $k^*$, and a central subspace, $U^*
\in$ Gr$(k^*,n)$ at the center of the minimum enclosing ball, that best
represents the data.
