*Published Paper*

**Inserted:** 24 mar 2004

**Last Updated:** 15 oct 2006

**Journal:** Calculus of Variations: Topics from the Mathematical Heritage of Ennio De Giorgi, D. Pallara (ed.), Quaderni di Matematica, Seconda Universit\`{a} di Napoli, Caserta

**Volume:** 14

**Pages:** 47-83

**Year:** 2004

**Abstract:**

The paper is concerned with problems of finding minima
among the set of all compact connected subsets $\Sigma\subset
\Omega$ ($\Omega\subset *R*^n$ being a given compact ambient set)
of the functional
$$
I(\Sigma):= \int_{\Omega} A(dist(x,\Sigma))\, d\varphi(x)
$$
subject to some penalization on the length ${\cal H}(\Sigma)$ of $\Sigma$,
where $A$: $*R*^+\to *R*^+$ is a given nondecreasing function and
$dist(x,\Sigma)$ stands for the distance between $x$ and $\Sigma$
defined with the use of the geodesic distance $d$ relative to $\Omega$
in the usual way, i.e.
$$
dist(x, \Sigma) := \inf_{{z\in\Sigma}} d(x,z).
$$
We provide various applications which are sources of similar problems,
and give some results concerning qualitative properties of minimizers
in the case $\Omega$ is a convex set (with $d$ a Euclidean distance).

**Keywords:**
Monge-Kantorovich problem, average distance functional, transportation network

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