Detecting product splittings of CAT(0) spaces

Let $X$ be a proper CAT($0$) space and $G$ a cocompact group of isometries of $X$ without fixed point at infinity. We prove that if $\partial X$ contains an invariant subset of circumradius $\pi/2$, then $X$ contains a quasi-dense, closed convex subspace that splits as a product. Adding the assumption that the $G$-action on $X$ is properly discontinuous, we give more conditions that are equivalent to a product splitting. In particular, this occurs if $\partial X$ contains a proper nonempty, closed, invariant, $\pi$-convex set in $\partial X$; or if some nonempty closed, invariant set in $\partial X$ intersects each round sphere $K \subset \partial X$ inside a proper subsphere of $K$.