Birkhoff's invariant and Thorne's Hoop Conjecture

I propose a sharp form of Thorne's hoop conjecture which relates Birkhoff's invariant $\beta$ for an outermost apparent horizon to its $ADM$ mass, $\beta \le 4 \pi M_{ADM}$. I prove the conjecture in the case of collapsing null shells and provide further evidence from exact rotating black hole solutions. Since $\beta$ is bounded below by the length $l$ of the shortest non-trivial geodesic lying in the apparent horizon, the conjecture implies $l \le 4 \pi M_{ADM}$. The Penrose conjecture, $\sqrt{\pi A} \le 4 \pi M_{ADM}$, and Pu's theorem imply this latter consequence for horizons admitting an antipodal isometry. Quite generally, Penrose's inequality and Berger's isembolic inequality, $\sqrt{\pi A} \ge {2\over\sqrt\pi} i$, where $i$ is the injectivity radius, imply $4c \le 2 i \le 4 \pi M_{ADM}$, where $c$ is the convexity radius.