  The joint convexity of the map $(X,A) \mapsto X^* A^{-1} X$, an integral
representation of operator convex functions, and an observation of Ando are
used to obtain a simple proof of both the joint convexity of relative entropy
and a trace convexity result of Lieb. The latter was the key ingredient in the
original proof of the strong subadditivity of quantum entropy.
