On Divergence-based Distance Functions for Multiply-connected Domains

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Given a finitely-connected bounded planar 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 our case are the Poisson kernels of the domain at x and at y. We prove that for the χ²-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 extremum in Ω, namely at y where D vanishes. Thus D can be used to trace a gradient-descent path within Ω from x to y by following ∇ₓ D(x,y), which has significant computational advantages over tracing the gradient of G. This result can be used for robotic path-planning in complex geometric environments.