  We consider two non-convex formulations for computing the optimal constant in
the Brascamp-Lieb inequality corresponding to a given datum, and show that they
are geodesically log-concave on the manifold of positive definite matrices
endowed with the Riemannian metric corresponding to the Hessian of the
log-determinant function. The first formulation is present in the work of Lieb
and the second is inspired by the work of Bennett et al. Recent works of Garg
et al.and Allen-Zhu et al. also imply a geodesically log-concave formulation of
the Brascamp-Lieb constant through a reduction to the operator scaling problem.
However, the dimension of the arising optimization problem in their reduction
depends exponentially on the number of bits needed to describe the
Brascamp-Lieb datum. The formulations presented here have dimensions that are
polynomial in the bit complexity of the input datum.
