On the distribution of distances in homogeneous compact metric spaces

We provide a simple proof that in any homogeneous, compact metric space of diameter $D$, if one finds the average distance $A$ achieved in $X$ with respect to some isometry invariant Borel probability measure, then $$\frac{D}{2} \leq A \leq D.$$ This result applies equally to vertex-transitive graphs and to compact, connected, homogeneous Riemannian manifolds. We then classify the cases where one of the extremes occurs. In particular any homogeneous compact metric space where $A=\frac{D}{2}$ possesses a strict antipodal property which implies in particular that the distribution of distances in $X$ is symmetric about $\frac{D}{2}$ which is hence both mean and median of the distribution. In particular, we show that the only closed, connected, positive-dimensional Riemannian manifolds with this strict antipodal property are spheres.