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We develop a framework for studying the well-known universal term in theRenyi entropy for an arbitrary entangling region in four-dimensional conformalfield theories that are holographically dual to gravitational theories. Theshape dependence of the Renyi entropy $S_n$ is described by two coefficients:$f_b(n)$ for traceless extrinsic curvature deformations and $f_c(n)$ for Weyltensor deformations. We provide the first calculation of the coefficient$f_b(n)$ in interacting theories by relating it to the stress tensor one-pointfunction in a deformed hyperboloid background. The latter is then determined bya straightforward holographic calculation. Our results show that a previousconjecture $f_b(n) = f_c(n)$, motivated by surprising evidence from a varietyof free field theories and studies of conical defects, fails holographically.

Shape Dependence of Holographic Renyi Entropy in Conformal Field Theories