  Given a finitely-connected bounded planar domain $\Omega$, it is possible to
define a {\it divergence distance} $D(x,y)$ from $x\in\Omega$ to $y\in\Omega$,
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
$\chi^2$-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 $\Omega$, namely at $y$ where $D$ vanishes. Thus $D$ can be used to
trace a gradient-descent path within~$\Omega$ from $x$ to $y$ by following
$\nabla_x D(x,y)$, which has significant computational advantages over tracing
the gradient of $G$. This result can be used for robotic path-planning in
complex geometric environments.
