  Benguria and Loss have conjectured that, amongst all smooth closed curves of
length $2\pi$ in the plane, the lowest possible eigenvalue of the operator
$L=-\Delta+\kappa^2$ was one. They observed that this value was achieved on a
two-parameter family, $\mathcal{O}$, of geometrically distinct ovals containing
the round circle and collapsing to a multiplicity-two line segment. We
characterize the curves in $\mathcal{O}$ as absolute minima of two related
geometric functionals. We also discuss a connection with projective
differential geometry and use it to explain the natural symmetries of all three
problems.
