  Let $U(\boldsymbol r),\boldsymbol r\in\Omega\subset \mathbb R^2$ be a
harmonic function that solves an exterior Dirichlet problem. If all the level
sets of $U(\boldsymbol r),\boldsymbol r\in\Omega$ are smooth Jordan curves,
then there are several geometric inequalities that correlate the curvature
$\kappa(\boldsymbol r) $ with the magnitude of gradient $ |\nabla U(\boldsymbol
r)|$ on each level set ("equipotential curve"). One of such inequalities is $
\langle [\kappa(\boldsymbol r)-\langle\kappa(\boldsymbol r)\rangle][|\nabla
U(\boldsymbol r)|-\langle |\nabla U(\boldsymbol r)|\rangle]\rangle\geq0$, where
$ \langle \cdot\rangle$ denotes average over a level set, weighted by the arc
length of the Jordan curve. We prove such a geometric inequality by
constructing an entropy for each level set $U(\boldsymbol r)=\varphi $, and
showing that such an entropy is convex in $\varphi$. The geometric inequality
for $\kappa(\boldsymbol r) $ and $ |\nabla U(\boldsymbol r)|$ then follows from
the convexity and monotonicity of our entropy formula. A few other geometric
relations for equipotential curves are also built on a convexity argument.
