Illumination bodies on Riemannian manifolds

We prove a generalization of Werner's asymptotic formula for the volume of the illumination body of a convex body, which holds on Riemannian manifolds with Ricci curvature bounded from below. The $\delta$-illumination body of a subset of a Riemannian manifold is defined to be the set of all points such that the union of all minimizing geodesic segments joining the point to the set has volume at most $\delta$.