Riemannian barycentres and geodesic convexity

Consider a closed subset of a complete Riemannian manifold, such that all g eodesics with end-points in the subset are contained in the subset and the subset has boundary of codimension one. Is it the case that Riemannian bary centres of probability measures supported by the subset must also lie in th e subset? It is shown that this is the case for 2-manifolds but not the eas e in higher dimensions: a counterexample is constructed which is a conforma lly-Euclidean 3-manifold, for which geodesics never self-intersect and inde ed cannot turn by too much (so small geodesic balls satisfy a geodesic conv exity condition), but is such that a probability measure concentrated on a single point has a barycentre at another point.