  The entanglement entropy in three-dimensional conformal field theories (CFTs)
receives a logarithmic contribution characterized by a regulator-independent
function $a(\theta)$ when the entangling surface contains a sharp corner with
opening angle $\theta$. In the limit of a smooth surface
($\theta\rightarrow\pi$), this corner contribution vanishes as
$a(\theta)=\sigma\,(\theta-\pi)^2$. In arXiv:1505.04804, we provided evidence
for the conjecture that for any $d=3$ CFT, this corner coefficient $\sigma$ is
determined by $C_T$, the coefficient appearing in the two-point function of the
stress tensor. Here, we argue that this is a particular instance of a much more
general relation connecting the analogous corner coefficient $\sigma_n$
appearing in the $n$th R\'enyi entropy and the scaling dimension $h_n$ of the
corresponding twist operator. In particular, we find the simple relation
$h_n/\sigma_n=(n-1)\pi$. We show how it reduces to our previous result as
$n\rightarrow 1$, and explicitly check its validity for free scalars and
fermions. With this new relation, we show that as $n\rightarrow 0$, $\sigma_n$
yields the coefficient of the thermal entropy, $c_S$. We also reveal a
surprising duality relating the corner coefficients of the scalar and the
fermion. Further, we use our result to predict $\sigma_n$ for holographic CFTs
dual to four-dimensional Einstein gravity. Our findings generalize to other
dimensions, and we emphasize the connection to the interval R\'enyi entropies
of $d=2$ CFTs.
