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Some results on the shape dependence of entanglement and Rényi entropies

We study how the universal contribution to entanglement entropy in a conformal field theory depends on the entangling region. We show that for a deformed sphere the variation of the universal contribution is quadratic in the deformation amplitude. We generalize these results for Rényi entropies. We obtain an explicit expression for the second order variation of entanglement entropy in the case of a deformed circle in a three dimensional CFT with a gravity dual. For the same system, we also consider an elliptic entangling region and determine numerically the entanglement entropy as a function of the aspect ratio of the ellipse. Based on these three-dimensional results and Solodukhin's formula in four dimensions, we conjecture that the sphere minimizes the universal contribution to entanglement entropy in all dimensions.