A characterization of spaces of constant curvature by minimum covering radius of triangles

In this paper, we prove that if in a Riemannian manifold, the minimum covering radius of a point triple of small diameter depends only on the geodesic distances between the points, then the manifold must be of constant curvature. This implies that if in a complete connected Riemannian manifold, the volume of the intersection of three small geodesic balls of equal radii depends only on the distances between the centers and the radius, then it is one of the simply connected spaces of constant curvature. This generalizes an earlier result of the first author and D. Kunszenti-Kovács (2010).