  A famous result of Lieb establishes that the map $(A,B) \mapsto
\text{tr}\left[K^* A^{1-t} K B^t\right]$ is jointly concave in the pair $(A,B)$
of positive definite matrices, where $K$ is a fixed matrix and $t \in [0,1]$.
In this paper we show that Lieb's function admits an explicit semidefinite
programming formulation for any rational $t \in [0,1]$. Our construction makes
use of a semidefinite formulation of weighted matrix geometric means. We
provide an implementation of our constructions in Matlab.
