  The mild form of the Weak Gravity Conjecture states that quantum or
higher-derivative corrections should decrease the mass of large extremal
charged black holes at fixed charge. This allows extremal black holes to decay,
unless protected by a symmetry (such as supersymmetry). We reformulate this
conjecture as an integrated condition on the effective stress tensor capturing
the effect of quantum or higher-derivative corrections. In addition to charged
black holes, we also consider rotating BTZ black holes and show that this
condition is satisfied as a consequence of the $c$-theorem, proving a spinning
version of the Weak Gravity Conjecture. We also apply our results to a
five-dimensional boosted black string with higher-derivative corrections. The
boosted black string has a $\text{BTZ}\times S^2$ near-horizon geometry and,
after Kaluza-Klein reduction, describes a four-dimensional charged black hole.
Combining the spinning and charged Weak Gravity Conjecture we obtain positivity
bounds on the five-dimensional Wilson coefficients that are stronger than those
obtained from charged black holes alone.
