*Published Paper*

**Inserted:** 5 mar 2009

**Last Updated:** 8 nov 2016

**Journal:** SIAM J. Math. Anal.

**Volume:** 42

**Number:** 3

**Pages:** 1179-1217

**Year:** 2010

**Doi:** 10.1137/100782693

**Abstract:**

In this paper we consider the class a functionals (introduced by Brancolini, Buttazzo, and Santambrogio) $\mathcal{G}_{r,p}(\gamma)$ defined on Lipschitz curves $\gamma$ valued in the $p$-Wasserstein space. The problem considered is the following: given a measure $\mu$, give conditions in order to assure the existence a curve $\gamma$ such that $\gamma(0) = \mu$, $\gamma(1) = \delta_{x_0}$, and $\mathcal{G}_{r,p}(\gamma) < +\infty$.

To this end, new estimates on $\mathcal{G}_{r,p}(\mu)$ are given and a notion of dimension of a measure (called *path dimension*) is introduced: the path dimension specifies the values of the parameters $(r,p)$ for which the answer to the previous reachability problem is positive. Finally, we compare the path dimension with other known dimensions.

**Keywords:**
Path functionals, optimal transportation problems, branched transportation

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