  We introduce a new geometric approach for the homogenization and inverse
homogenization of the divergence form elliptic operator with rough conductivity
coefficients $\sigma(x)$ in dimension two. We show that conductivity
coefficients are in one-to-one correspondence with divergence-free matrices and
convex functions $s(x)$ over the domain $\Omega$. Although homogenization is a
non-linear and non-injective operator when applied directly to conductivity
coefficients, homogenization becomes a linear interpolation operator over
triangulations of $\Omega$ when re-expressed using convex functions, and is a
volume averaging operator when re-expressed with divergence-free matrices.
Using optimal weighted Delaunay triangulations for linearly interpolating
convex functions, we obtain an optimally robust homogenization algorithm for
arbitrary rough coefficients. Next, we consider inverse homogenization and show
how to decompose it into a linear ill-posed problem and a well-posed non-linear
problem. We apply this new geometric approach to Electrical Impedance
Tomography (EIT). It is known that the EIT problem admits at most one isotropic
solution. If an isotropic solution exists, we show how to compute it from any
conductivity having the same boundary Dirichlet-to-Neumann map. It is known
that the EIT problem admits a unique (stable with respect to $G$-convergence)
solution in the space of divergence-free matrices. As such we suggest that the
space of convex functions is the natural space in which to parameterize
solutions of the EIT problem.
