That harmonic functions would satisfy the mean value property for infinitesimally small circles is intuitively somewhat clear. But that this property should persist on circles of arbitrary radius is quite miraculous. To recall, if is harmonic on a domain and one considers a disk , then
One might wonder whether such a global property occurs for other shapes, say ellipses. The answer is no.
Fact: Suppose is open and has finite area , , and
for every integrable harmonic function on . Then is a disk centered at .
To see why, pick a point that is closest to . Then play with the function