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We study how the universal contribution to entanglement entropy in aconformal field theory depends on the entangling region. We show that for adeformed sphere the variation of the universal contribution is quadratic in thedeformation amplitude. We generalize these results for R\'enyi entropies. Weobtain an explicit expression for the second order variation of entanglemententropy in the case of a deformed circle in a three dimensional CFT with agravity dual. For the same system, we also consider an elliptic entanglingregion and determine numerically the entanglement entropy as a function of theaspect ratio of the ellipse. Based on these three-dimensional results andSolodukhin's formula in four dimensions, we conjecture that the sphereminimizes the universal contribution to entanglement entropy in all dimensions.

Some results on the shape dependence of entanglement and R\'enyi entropies