Let $f$ be a continuous function defined on $\mathbb{R^n}$. It is well known that both the spherical mean value property (MVP) of $f$, i.e. $$f(x)=\frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}f,\ \forall x\in\mathbb{R^n}, r>0$$ and the ball MVP, i.e. $$f(x)=\frac{1}{|B(x,r)|}\int_{B(x,r)}f,\ \forall x\in\mathbb{R^n},r>0$$ imply that $f$ is harmonic.

Note that in the definitions we require the redius $r$ to run over all the positive numbers. Out of curiosity I tried to find non-harmonic functions which satysfy the MVPs *only* for $r=1$. I did some search and found a remarkable fact called Delsarte's two-radius theorem saying that the spherical MVP with *two* fixed radii is enough to imply harmonicity of $f$. But for the $1$-radius MVP I haven't found any statement.

In the case $n=1$ examples have been found nicely in this M.SE post. But it is still unclear to me how to construct similar examples in higher dimensions. Any comments would be appreciated!