We consider constrained optimization of geodesically convex objectives over geodesically convex subsets of Riemannian manifolds. We address these constraints via a Riemannian Frank-Wolfe (RFW) approach, that o ers to be promising due to its “projection free” nature. First, we prove an abstract convergence result for RFW; then we specialize it to the manifold of positive de nite matrices, for the speci c task of computing the Riemannian centroid (also known as Karcher mean). This specialization relies crucially on a “log-linear oracle,” a subroutine key to implementing RFW – remarkably, this oracle is seen to admit a closed form solution, which may be of independent interest. We discuss two other variations, including a non- convex Euclidean Frank-Wolfe (EFW) method, as well a setting under which RFW attains a linear rate of con- vergence. Experiments against recently published methods for the Riemannian centroid highlight the competitiveness of RFW.