Let $M$ be a simply connected space form (i.e. $\mathbb R^n$, sphere, or hyperbolic space) and $B$ be a ball in $M$. Let $\phi$ be the first Laplacian eigenfunction on $B$, with respect to the Dirichlet boundary condition $\phi=0$ on $\partial B$.

In Cheng's remarkable paper "Eigenvalue comparison theorems and its geometric applications", it is remarked that (p. 290) "since all simply connected space forms are two-point homogeneous, $\phi$ is a radial function". I think two-point homogeneity means that there always exists an isometry mapping one pair of points to any other pair of points which have the same distance apart (is it?). I don't understand how this condition is used.

I think the reason why $\phi$ is radial is that we can apply the spherical mean of it to obtain $\widetilde \phi$, which is also an eigenfunction as rotation preserves the metric. If $\widetilde \phi\ne \phi$, then $h=\phi-\widetilde \phi$ is a first eigenfunction that changes sign, which is a contradiction. But I don't see where the "two-point homogeneous condition" is used. (Is there an alternate proof?)

If my argument is correct it should also apply to balls centered at 0 of the warped product space $dr^2+f(r)^2 d\theta^2$. Is this well-known? If so, is there a quick reference?