A presentation of the average distance minimizing problem

We talk about the following minimization problem
$$ \min F(\Sigma):=\int_\Omega d(x,\Sigma)\,\mathrm d\mu(x), $$
where $\Omega$ is an open subset of $\mathbb R^2$, $\mu$ is a probability measure and where the minimum is taken over all the sets $\Sigma\subset\overline\Omega$ such that $\Sigma$ is compact, connected, and $\mathcal H^1(\Sigma)\leq\alpha_0$ for a given positive constant $\alpha_0$.