  In this note, we study an optimal transportation problem arising in density
functional theory. We derive an upper bound on the semi-classical
Hohenberg-Kohn functional derived by Cotar, Friesecke and Kl\"{u}ppelberg
(2012) which can be computed in a straightforward way for a given single
particle density. This complements a lower bound derived by the aforementioned
authors. We also show that for radially symmetric densities the optimal
transportation problem arising in the semi-classical Hohenberg-Kohn functional
can be reduced to a 1-dimensional problem. This yields a simple new proof of
the explicit solution to the optimal transport problem for two particles found
by Cotar, Friesecke and Kl\"{u}ppelberg (2012). For more particles, we use our
result to demonstrate two new qualitative facts: first, that the solution can
concentrate on higher dimensional submanifolds and second that the solution can
be non-unique, even with an additional symmetry constraint imposed.
