Does negative type characterize the round sphere?

We discuss the measure-theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact Riemannian manifold with symmetry ratio 1 must be a round sphere was put forward by the author in 2004. We resolve this conjecture in the class of Riemannian symmetric spaces by showing that a Riemannian manifold with symmetry ratio 1 must be of negative type and that the only compact Riemannian symmetric spaces of negative type are the round spheres.