  The Lieb-Oxford bound, a nontrivial inequality for the indirect part of the
many-body Coulomb repulsion in an electronic system, plays an important role in
the construction of approximations in density functional theory. Using the
wavefunction for strictly-correlated electrons of a given density, we turn the
search over wavefunctions appearing in the original bound into a more
manageable search over electron densities. This allows us to challenge the
bound in a systematic way. We find that a maximizing density for the bound, if
it exists, must have compact support. We also find that, at least for particle
numbers $N\le 60$, a uniform density profile is not the most challenging for
the bound. With our construction we improve the bound for $N=2$ electrons that
was originally found by Lieb and Oxford, we give a new lower bound to the
constant appearing in the Lieb-Oxford inequality valid for any $N$, and we
provide an improved upper bound for the low-density uniform electron gas
indirect energy.
