*Preprint*

**Inserted:** 22 feb 2008

**Year:** 2008

**Abstract:**

We construct some examples of explicit solutions to the problem
$$
\min_{\gamma} \int_{\Omega} d_{\gamma}(x)\,dx
$$
where the minimum is over all connected compact sets $\gamma\subset\overline{\Omega}\subset R^2$ of
prescribed one-dimensional Hausdorff measure. More precisely
we show that, if $\gamma$ is a $C^{1,1}$ curve of length $l$
with curvature bounded by $1/R$, $l\leq \pi R$ and $\varepsilon\leq R$,
then $\gamma$ is a solution to the above problem with
$\Omega$ being the $\varepsilon$-neighbourhood of $\gamma$. In
particular, $C^{1,1}$ regularity is optimal for this problem.

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