Disks globally maximize the entanglement entropy in $2+1$ dimensions

The entanglement entropy corresponding to a smooth region in general three-dimensional CFTs contains a constant universal term, $-F \subset S_{\text{EE}}$. For a disk region, $F|_{\rm disk}\equiv F_0$ coincides with the free energy on $\mathbb{S}^3$ and provides an RG-monotone for general theories. As opposed to the analogous quantity in four dimensions, the value of $F$ generally depends in a complicated (and non-local) way on the geometry of the region and the theory under consideration. For small geometric deformations of the disk in general CFTs as well as for arbitrary regions in holographic theories, it has been argued that $F$ is precisely minimized by disks. Here, we argue that $F$ is globally minimized by disks with respect to arbitrary regions and for general theories. The proof makes use of the strong subadditivity of entanglement entropy and the geometric fact that one can always place an osculating circle within a given smooth entangling region. For topologically non-trivial entangling regions with $n_B$ boundaries, the general bound can be improved to $F \geq n_B F_0$. In addition, we provide accurate approximations to $F$ valid for general CFTs in the case of elliptic regions for arbitrary values of the eccentricity which we check against lattice calculations for free fields. We also evaluate $F$ numerically for more general shapes in the so-called "Extensive Mutual Information model", verifying the general bound.