  In this paper, we consider the electrostatic Born-Infeld model
\begin{equation*} \tag{$\mathcal{BI}$} \left\{ \begin{array}{rcll}
-\operatorname{div}\left(\displaystyle\frac{\nabla \phi}{\sqrt{1-|\nabla
\phi|^2}}\right)&=& \rho & \hbox{in }\mathbb{R}^N, \\[6mm]
\displaystyle\lim_{|x|\to \infty}\phi(x)&=& 0 \end{array} \right.
\end{equation*} where $\rho$ is a charge distribution on the boundary of a
bounded domain $\Omega\subset \mathbb{R}^N$. We are interested in its
equilibrium measures, i.e. charge distributions which minimize the
electrostatic energy of the corresponding potential among all possible
distributions with fixed total charge. We prove existence of equilibrium
measures and we show that the corresponding equilibrium potential is unique and
constant in $\overline \Omega$. Furthermore, for smooth domains, we obtain the
uniqueness of the equilibrium measure, we give its precise expression, and we
verify that the equilibrium potential solves ($\mathcal{BI}$). Finally we
characterize balls in $\mathbb{R}^N$ as the unique sets among all bounded
$C^{2,\alpha}$-domains $\Omega$ for which the equilibrium distribution is a
constant multiple of the surface measure on $\partial\Omega$. The same results
are obtained also for Taylor approximations of the electrostatic energy.
