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

On Geodesically Convex Formulations for the Brascamp-Lieb Constant