  We introduce a notion of halfspace for Hadamard manifolds that is natural in
the context of convex optimization. For this notion of halfspace, we generalize
a classic result of Gr\"unbaum, which itself is a corollary of Helly's theorem.
Namely, given a probability distribution on the manifold, there is a point for
which all halfspaces based at this point have at least $\frac{1}{n+1}$ of the
mass. As an application, the gradient oracle complexity of convex optimization
is polynomial in the parameters defining the problem.
