  We quantify conditions that ensure that a signed measure on a Riemannian
manifold has a well defined centre of mass. We then use this result to quantify
the extent of a neighbourhood on which the Riemannian barycentric coordinates
of a set of $n+1$ points on an $n$-manifold provide a true coordinate chart,
i.e., the barycentric coordinates provide a diffeomorphism between a
neighbourhood of a Euclidean simplex, and a neighbourhood containing the points
on the manifold.
