  The Friedland-Hayman inequality is a sharp inequality concerning the growth
rates of homogeneous, harmonic functions with Dirichlet boundary conditions on
complementary cones dividing Euclidean space into two parts. In this paper, we
prove an analogous inequality in which one divides a convex cone into two
parts, placing Neumann conditions on the boundary of the convex cone, and
Dirichlet conditions on the interface. This analogous inequality was already
proved by us jointly with Sarah Raynor. Here we present a new proof that
permits us to characterize the case of equality. In keeping with the two-phase
free boundary theory introduced by Alt, Caffarelli, and Friedman, such an
improvement can be expected to yield further regularity in free boundary
problems.
