  In this note we study estimates from below of the single radius spherical
discrepancy in the setting of compact two-point homogeneous spaces. Namely,
given a $d$-dimensional manifold $\mathcal M$ endowed with a distance $\rho$ so
that $(\mathcal M, \rho)$ is a two-point homogeneous space and with the
Riemannian measure $\mu$, we provide conditions on $r$ such that if $D_r$
denotes the discrepancy of the ball of radius $r$, then, for an absolute
constant $C>0$ and for every set of points $\{x_j\}_{j=1}^N$, one has
$\int_{\mathcal M} |D_{r}(x)|^2\, d\mu(x)\geqslant C N^{-1-\frac1d}$. The
conditions on $r$ that we have depend on the dimension $d$ of the manifold and
cannot be achieved when $d \equiv 1 \ ( \operatorname{mod}4)$. Nonetheless, we
prove a weaker estimate for such dimensions as well.
