TITLE: Does the pointwise mean value property imply harmonicity?
QUESTION [7 upvotes]: Assume $u:\Omega\subset\mathbb{R}^d\to\mathbb{R}$ is continuous and satisfies the property:
for every $x\in \mathbb{\Omega}$ there is $r_x>0$ such that
$$
u(x)=\frac{1}{|B(x,r_x)|}\int_{B(x,r_x)} u(y)\, dy,\hskip .1in\text{(Discrete Mean-value property).}\tag{1}\label{1}$$
What can we say about the harmonicity of $u$? which kind of harmonicity do we have?
Recall that if \eqref{1} holds independently of $r_x$ (that is for every $r>0$ such that $B(x,r)\subset \Omega$), then $u$ is harmonic.

REPLY [8 votes]: This question was addressed by Hansen and Nadirashvili in a series of papers, see, for example:
MR1315353
Hansen, W., Nadirashvili, N.,
On Veech's conjecture for harmonic functions.
Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 1, 137–153.
and references there.