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On Geodesically Convex Formulations for the Brascamp-Lieb Constant Sra, Suvrit Vishnoi, Nisheeth K. Yildiz, Ozan Geodesic convexity positive definite cone geodesics Brascamp-Lieb constant 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 [Lieb, 1990] and the second is new and inspired by the work of Bennett et al. [Bennett et al., 2008]. Recent work of Garg et al. [Ankit Garg et al., 2017] also implies 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Suvrit Sra and Nisheeth K. Vishnoi and Ozan Yildiz 2018 Is Part Of LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.APPROX-RANDOM.2018.25 urn:nbn:de:0030-drops-94296 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.25 eng https://creativecommons.org/licenses/by/3.0/legalcode