*Published Paper*

**Inserted:** 15 feb 2004

**Last Updated:** 11 jun 2012

**Journal:** Calc. Var. Partial Diff. Equations

**Volume:** 27

**Number:** 3

**Pages:** 287-309

**Year:** 2006

**Abstract:**

Consider the class of closed connected sets $\Sigma\subset R^n$ satisfying length constraint ${\cal H}(\Sigma)\leq l$ with given $l>0$. The paper is concerned with the properties of minimizers of the uniform distance $F_M$ of $\Sigma$ to a given compact set $M\subset R^n$, \[ F_M(\Sigma):= \max_{y\in M}dist(y,\Sigma), \] where $dist(y, \Sigma)$ stands for the distance between $y$ and $\Sigma$. The paper deals with the planar case $n=2$. In this case it is proven that the minimizers (apart trivial cases) cannot contain closed loops. Further, some mild regularity properties as well as structure of minimizers is studied.

**Keywords:**
urban planning, transport network, multicenter problem, multimedian problem

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