It is a well known fact that a harmonic function in $\mathbb{R}^n$ has the mean value property, namely: the average value of a harmonic function at the border of any (euclidean) ball is equal to the value at its center (and hence, also to the average value in the interior of the ball).

We say a function $f$ is subharmonic if, for any ball $B(x,r)$ such that $f\leq u$ in $\partial B(x,r)$ for some harmonic function $u$, then $f\leq u$. Reversing the inequalities, we get the definition of superharmonic functions. From the mean value property and these definitions it is easy to derive that sub/superharmonic functions' averages in the border of a ball $B(x,r)$ are monotone with respect to the radius of the ball. Given a subharmonic function $u$ and two radius $r>s$, you may take a harmonic function $f$ such that $u=f$ in the ball $B(x,r)$. As the average of $f$ equals in $B(x,r)$ equals the average in $B(x,s)$, which is greater than the average of $u$, the monotonicity is proved.

Now, my question is whether this property can be derived also for balls which are not euclidean, in particular using any $p$-metric for $1\leq p\leq\infty$. There are some questions in here regarding the mean value property for other balls or sets (see this one and this one), but the answers are not entirely satisfactory. As far as I know, the main obstacle to adapt a standard proof of the mean value property (see Salsa's book on PDEs, for instance) to other balls is that it's not possible to apply the divergence theorem anymore. So a proof of the mean value property for other balls should be sufficient to answer this question. Otherwise, there might be some other way to prove the monotonicity outright which would also be very welcome.



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