Let $f:\mathbb{R}^n\to\mathbb{R}$ be a harmonic function. The mean value property states that for all $r>0$, the average of $f$ over a ball $B(x,r)$ is equal to the average of $f$ over a $\partial B(x,r)$, and both of them are equal to $f(x)$.
Suppose that $f$ satisfies a weaker mean value property: for all $x\in \mathbb{R}^n$, $f(x)$ is equal to the average of $f$ over $\partial B(x,1)$.
Is the weaker property equivalent to the original one? or can you give a counter example?
Does this hold in greater generality (say, of Lie groups)?