Question:

# When the value of a function in a point is equal to its integral average over the point's neighborhood?

Carter: 2 days ago

It is well-known that the harmonic functions have this remarkable Averaging Property: if $f$ is harmonic in a domain $U \subset R^n$, then, for any point $x \in U$, $f(x)$ is equal to the integral average of $f$ over any ball centered at $x$ and lying in $U$.

Here a ball is understood with respect to the Euclidean distance -- generated by the norm $\sqrt{\sum_{i=1}^{n} x_i^2}.$ But one can consider the "balls" with respect to other "classical" distances in $R^n$ -- generated by such norms as $\max(|x_1|,|x_2|,...,|x_n| )$ or $\sum_{i=1}^{n} |x_i|$. I have done some research investigating functions satisfying this Averaging Property with respect to the "balls"'corresponding to the distances generated by other norms in $R^n$ (such as the two norms listed above). However I suspect that some studies on this topic have been done already. I would be most grateful if somebody could inform me about publications on this topic.