  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 $x_0$ in the domain, there exists a
nested family of sets $\{ D_r(x_0) \}$ where the average over any of those sets
is related to the value of the function at $x_0.$ Although it is known that the
$\{ D_r(x_0) \}$ are nested and are comparable to balls in the sense that there
exists $c, C$ depending only on $L$ such that $B_{cr}(x_0) \subset D_r(x_0)
\subset B_{Cr}(x_0)$ for all $r > 0$ and $x_0$ 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.
