I would appreciate any insights on this matter;
Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic function at the center of a sphere is the average of values of this function over the entire sphere.
Now suppose I have an ellipsoid which is very skewed in one direction. Can one write the mean value property with respect to a different measure? Furthermore are there any inequalities relating the average of the harmonic function over the whole ellipsoid and the value at the center?