One-dimensional projective structures, convex curves and the ovals of Benguria & Loss

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.