Mean value property with fixed radius 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!
 A: It is a theorem of Hansen and Nadirashvili: W. Hansen, N. Nadirashvili, "Littlewood's one circle problem" J. London Math. Soc. , 50 (1994) pp. 349–360 that one radius (which is allowed to be a function of the point) is not enough for the spherical MVP for functions defined in domains in $\mathbb{C}$ -- this is still open in dimension > 2, while it IS enough for the ball MVP (W. Hansen, N. Nadirashvili, "A converse to the mean value theorem for harmonic functions" Acta Math. , 171 (1993) pp. 139–163), in every dimension. If instead of a domain we take all of $\mathbb{C}$ the answer is YES, and the proof is elementary, see:
W. Hansen, A strong version of Liouville’s theorem, Am. Math. Mon. 115 (7) (2008) 583–595.
EDIT
To answer @George's comment: I was actually misled by Hansen's statement in a later paper. What he actually seems to show is that a positive bounded function satisfying the mean-value property for a single $r(x)$ at every $x,$ where $r(x) < | x| + M,$ for some $M$ is constant.
