On Divergence-based Distance Functions for Multiply-connected Domains

Given a finitely connected planar Jordan domain Ω, it is possible to define a divergence distance D(x, y) from x ∈ Ω to y ∈ Ω, which takes into account the complex geometry of the domain. This distance function is based on the concept of f -divergence, a distance measure traditionally used to measure the difference between two probability distributions. The relevant probability distributions in this case are the Poisson kernels of the domain at x and at y. We prove that for the χ-divergence distance, the gradient by x of D is opposite in direction to the gradient by x of G(x, y), the Green’s function with pole y. Since G is harmonic, this implies that D, like G, has a single minimum in Ω. Thus D can be used to trace a gradient-descent path within Ω from x to y by following ∇xD(x, y), which has computational advantages over tracing the gradient of G.