In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator L and at any point x0 in the domain, there exists a nested family of sets {Dr(x0)} where the average over any of those sets is related to the value of the function at x0. Although it is known that the {Dr(x0)} are nested and are comparable to balls in the sense that there exists c,C depending only on L such that Bcr(x0) ? Dr(x0) ? BCr(x0) for all r >?0 and x0 in the domain, otherwise their geometric and topological properties are largely unknown. In this paper we begin the study of these topics and we prove a few results about the geometry of these sets and give a couple of applications of the theorems.
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