By <cit.> we have $\xi\Ha\mres T+\tilde{\F}=\vartheta\Ha\mres M+\G$ with $M\subset\mathcal{C}$ countably $1$-rectifiable, $\vartheta\in L^1(\Ha\mres M;\R^n)$ and $\G\in\mathcal{M}^n(\mathcal{C})$ $\Ha$-diffuse. Define $\hat{\F}=\tilde{\F}\mres (\mathcal{C}\setminus S)$ and

\begin{equation*}
\hat{\xi}=\begin{cases}
\xi&\textup{on }T,\\
\vartheta&\textup{on }M\cap (S\setminus T),\\
\end{cases}
\end{equation*}

Then we have $\hat{\F}\in\mathcal{M}^n(\mathcal{C})$ with $\hat{\F}\mres S=0$ and $\hat{\xi}\in L^1(\Ha\mres S;\R^n)$ by definition. Additionally, we get

\begin{equation*}
\hat{\xi}\Ha\mres S+\hat{\F}=\xi\Ha\mres T+\vartheta\Ha\mres (M\cap (S\setminus T))+\hat{\F}=\xi\Ha\mres T+\tilde{\F}\mres (S\setminus T)+\tilde{\F}\mres (\mathcal{C}\setminus S)=\xi\Ha\mres T+\tilde{\F}.
\end{equation*}

Finally, we can estimate

\begin{align*}
\int_{S}b|\hat{\xi}|\,\mathrm{d}\Ha+a|\hat{\F}|(\mathcal{C} )&=\int_{T}b|\xi|\,\mathrm{d}\Ha+\int_{M\cap (S\setminus T)}b|\vartheta|\,\mathrm{d}\Ha+a|\hat{\F}|(\mathcal{C} )\leq \int_{T}b|\xi|\,\mathrm{d}\Ha+a|\vartheta\Ha\mres (M\cap (S\setminus T))|(\mathcal{C} )+a|\hat{\F}|(\mathcal{C} )\\
\end{align*}

which is the desired inequality.

$W_d(\mu_+,\mu_-)\leq B^{\mu_+,\mu_-}$: Initially, we show the inequality for the case $\inf_Sb>0$. Let $\delta>0$ and fix $\xi$ and $\tilde{\F}$ like in the Beckmann problem. We define measures by $\G_N=\xi\Ha\mres (S\setminus S^N)+\tilde{\F}$. $|\xi|$ is integrable with respect to $\Ha\mres S$. Consequently, we have

\begin{equation*}
|\G_N-\tilde{\F}|(\mathcal{C} )=\int_{S\setminus S_N}|\xi|\,\mathrm{d}\Ha\to 0
\end{equation*}

for $N\to\infty$. Thus, we can choose $N$ sufficiently large such that $|\G_N-\tilde{\F}|(\mathcal{C})<\delta/(2a)$. By <ref> and <ref> there exist optimal transport plans $\pi_N\in\Pi(\mu_+,\mu_-)$ such that $W_{d^N}(\mu_+,\mu_-)=\langle d^N,\pi_N\rangle\to\langle d,\pi\rangle$ up to a subsequence for some $\pi\in\Pi(\mu_+,\mu_-)$. From now on, we will only consider the corresponding subsequence. We choose $N$ sufficiently large such that $|\langle d^N,\pi_N\rangle-\langle d,\pi\rangle|<\delta/2$. Using $\textup{div}(\xi\Ha\mres  S^N+\G_N)=\textup{div}(\xi\Ha\mres S+\tilde{\F})=\mu_+-\mu_-$, <ref> and <ref> we can estimate

\begin{align*}
\int_Sb|\xi|\,\mathrm{d}\Ha+a|\tilde{\F}|(\mathcal{C} )&\geq \int_{S_N}b|\xi|\,\mathrm{d}\Ha+a|\G_i|(\mathcal{C} )-\delta/2\geq \inf_{\substack{\Xi\in\mathcal{M}^n(S^N)\\ \Sigma\in\mathcal{M}^n(\mathcal{C})}} \int_{S^N}b\,\mathrm{d}|\Xi|+a|\Sigma|(\mathcal{C} )+\iota_{\{ \mu_+-\mu_- \}}(\textup{div}(\Xi+\Sigma))-\delta/2\\
&=\sup_{\varphi\in D_N}\int_{\mathcal{C} }\varphi\,\mathrm{d}(\mu_+-\mu_-)-\delta/2=W_{d^N}(\mu_+,\mu_-)-\delta/2
=\langle d^N,\pi_N\rangle-\delta/2
\geq \langle d,\pi\rangle-\delta
\\&\geq W_d(\mu_+,\mu_-)-\delta,
\end{align*}

where $D_N=\{ \varphi\in C^1(\mathcal{C})\,|\,|\nabla\varphi|\leq b\textup{ on }S^N,|\nabla\varphi|\leq a\textup{ in }\mathcal{C} \}$. Letting $\delta\to 0$ yields the desired result. Now we concentrate on the case $\inf_Sb=0$. Define $b_\lambda=\max\{\lambda,b\}$ for $\lambda\in (0,a]$. Then we obtain

\begin{equation*}
\int_Sb_\lambda|\xi|\,\mathrm{d}\Ha+a|\tilde{\F}|(\mathcal{C} )\geq W_d(\mu_+,\mu_-).
\end{equation*}

Moreover, we have

\begin{equation*}
\int_Sb_\lambda|\xi|\,\mathrm{d}\Ha\leq a\int_S|\xi|\,\mathrm{d}\Ha<\infty
\end{equation*}

and therefore (monotone convergence)

\begin{equation*}
\int_Sb|\xi|\,\mathrm{d}\Ha+a|\tilde{\F}|(\mathcal{C} )\geq W_d(\mu_+,\mu_-)
\end{equation*}

by letting $\lambda\to 0$. Taking the infimum over $\xi$ and $\tilde{\F}$ yields the desired inequality.

$W_d(\mu_+,\mu_-)\geq B^{\mu_+,\mu_-}$: Firstly, we assume that $\inf_Sb>0$. By <ref> there exists an optimal transport plan $\pi\in\Pi(\mu_+,\mu_-)$ such that $W_d(\mu_+,\mu_-)=\langle d,\pi\rangle$. Let $\delta>0$ be arbitrary. Then it is straightforward to see $d^N\searrow d$ pointwise for $N\to\infty$ (cf. proof of <ref>). Using $\langle d^N,\pi\rangle\leq a\cdot\textup{diam}(\mathcal{C})<\infty$ the monotone convergence theorem implies the existence of $N=N(\delta)$ such that $|\langle d^N-d,\pi\rangle|\leq\delta$. Application of <ref>, <ref>, <ref> (refering to second equation) and <ref> (last inequality) yields

\begin{align*}
W_d(\mu_+,\mu_-)&=\langle d,\pi \rangle
\geq \langle d^N,\pi\rangle-\delta
\geq W_{d^N}(\mu_+,\mu_-)-\delta
=\inf_{\xi,\tilde{\F}}\int_{S^N}b|\xi|\,\mathrm{d}\Ha+a|\tilde{\F}|(\mathcal{C} )+\iota_{\{ \mu_+-\mu_-\}} (\xi\Ha\mres S^N+\F)-\delta\\
&\geq\inf_{\hat{\xi},\hat{\F}}\int_{S}b|\hat{\xi}|\,\mathrm{d}\Ha+a|\hat{\F}|(\mathcal{C} )+\iota_{\{ \mu_+-\mu_-\}} (\hat{\xi}\Ha\mres S+\hat{\F})-\delta,
\end{align*}

where the infima are taken over functions $\xi,\tilde{\F},\hat{\xi},\hat{\F}$ as in <ref> with $T=S^N$. Letting $\delta\to 0$ yields the desired inequality. For the case $\inf_Sb=0$ set $b_\lambda=\max\{\lambda,b \}$ for $\lambda\in (0,a]$. Let $\pi\in\Pi(\mu_+,\mu_-)$ and $\varepsilon>0$. It is straightforward to see that $d_{b_\lambda}\searrow d$ pointwise for $\lambda\to 0$. Again, by $\langle d_{b_\lambda},\pi\rangle<\infty$ we can apply the monotone convergence theorem and choose $\lambda$ sufficiently small such that

\begin{align*}
\langle d,\pi\rangle&\geq \langle d_{b_\lambda },\pi\rangle-\varepsilon
\geq \inf_{\hat{\xi},\hat{\F}}\int_{S}b_\lambda|\hat{\xi}|\,\mathrm{d}\Ha+a|\hat{\F}|(\mathcal{C} )+\iota_{\{ \mu_+-\mu_-\}} (\hat{\xi}\Ha\mres S+\hat{\F})-\varepsilon\\
&\geq \inf_{\hat{\xi},\hat{\F}}\int_{S}b|\hat{\xi}|\,\mathrm{d}\Ha+a|\hat{\F}|(\mathcal{C} )+\iota_{\{ \mu_+-\mu_-\}} (\hat{\xi}\Ha\mres S+\hat{\F})-\varepsilon
\end{align*}

with $\hat{\xi},\hat{\F}$ as above. Taking the infimum over $\pi $ and letting $\varepsilon\to 0$ we obtain the desired inequality.

§ BILEVEL FORMULATION OF THE BRANCHED TRANSPORT PROBLEM

Let $\mu_+, \mu_-$ be given probability measures on $\mathcal{B}(\mathcal{C})$. In this section we will prove <ref>: The branched transport problem (<ref>) of finding an optimal mass flux from $\mu_+$ to $\mu_-$ with respect to a (concave) transportation cost $\tau$ can be equivalently written as a generalized version of the urban planning problem (<ref>). We briefly recapitulate the setting from <ref>. In the urban planning problem one optimizes over countably $1$-rectifiable and Borel measurable networks $S\subset\mathcal{C}$ and lower semi-continuous friction coefficients $b:S\to [0,\infty)$ representing a street or pipe network,

\begin{equation*}
\inf_{S,b}\mathcal{U}^{c,\mu_+,\mu_-}[S,b]=\inf_{S,b}W_{d_{S,a,b}}(\mu_+,\mu_-)+\int_Sc(b)\dHa.
\end{equation*}

The optimization depends on a fixed, decreasing maintenance cost $c:\R\to[0,\infty]$, and the cost for motion outside the network is defined by $a=\inf c^{-1}(0)$. In the branched transport problem on the other hand the transportation cost is a concave function $\tau:[0,\infty)\to[0,\infty)$ with $\tau(0)=0$, and one looks for an optimal mass flux $\F\in\mathcal{DM}^n(\R^n)$ with $\textup{div}(\F)=\mu_+-\mu_-$,

\begin{equation*}
\inf_\F\mathcal{J}^{\tau,\mu_+,\mu_-}[\F],
\end{equation*}

where $\mathcal{J}^{\tau,\mu_+,\mu_-}$ is defined via relaxation of a discrete energy.
We extend $\tau$ to a function $\R\to[-\infty,\infty)$ by setting $\tau(m)=-\infty$ for $m<0$. Moreover, we set

\begin{equation*}
\tau'(0)=\lim_{m\searrow 0}\frac{\tau(m)}{m}\in[0,\infty].
\end{equation*}

We use the convex conjugate of $-\tau$ to define a maintenance cost for our generalized urban planning problem which will be shown to be equivalent to the branched transport problem for $\tau$,

\begin{equation*}
\varepsilon(v)=(-\tau)^*(-v)=\sup_{m\geq 0}\tau(m)-mv
\qquad\text{for }v\in\R.
\end{equation*}

We observe that by definition

\begin{equation*}
\tau'(0)=\inf\varepsilon^{-1}(0)=a.
\end{equation*}

The actual statement of <ref> to be shown in this section is

\begin{equation*}
\inf_{\F}\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]=\inf_{S,b}\mathcal{U}^{\varepsilon,\mu_+,\mu_-}[S,b].
\end{equation*}

In <ref> we will formulate an appropriate version of the branched transport problem that will later naturally lead to our Beckmann formulation of the Wasserstein distance from <ref>.
[inline]If Section 4.2 stays, uncomment lines referring to that.

The formula in <cit.> shows that the cost of an admissible mass flux for the branched transport problem can be decomposed into costs for the mass flux on a network $S$ like above and a diffuse part for the costs of the flux outside the network. In <ref> we will prove a dual formula for the first term which will be used to introduce the friction coefficient functions $b$ (<ref>). Based on these preparations,

In <ref> we then establish the equivalence between the branched transport and the urban planning problem, and we discuss the relation between minimizers of each.

§.§ Version of the branched transport problem

In this section we introduce the reformulation of the branched transport problem from <cit.> as a generalized Gilbert energy in order to prepare the equivalence proof in <ref>. Further, we highlight some properties of the variables appearing in the reformulation. We first note that it suffices to concentrate on mass fluxes with support in $\mathcal{C}$ (in fact, one may replace $\mathcal{C}$ by the convex hull of $\textup{supp}(\mu_+)\cup\textup{supp}(\mu_-)$, see <cit.>).

We have

\begin{equation*}
\inf_{\F\in\mathcal{DM}^n(\R^n) }\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]=\inf_{\F\in\mathcal{DM}^n(\mathcal{C})}\mathcal{J}^{\tau,\mu_+,\mu_-}[\F].
\end{equation*}

We will in this section work with the following expression for the branched transport cost.

Every $\F\in\mathcal{DM}^n(\mathcal{C} )$ satisfies $\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]<\infty$ if and only if

* $\textup{div}(\F)=\mu_+-\mu_-$,

* $\F=\xi\Ha \mres S+\F^\perp $ with countably $1$-rectifiable $S\subset\mathcal{C}$, $\xi :S\to\R^n$ $\Ha\mres S$-measurable (tangent to $S$ $\Ha$-almost everywhere) and $\F^\perp $ singular with respect to $\Ha\mres R$ for any countably $1$-rectifiable set $R\subset\mathcal{C} $.

Assume that $\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]<\infty$ and $\F=\xi\Ha \mres S+\F^\perp$ as above. Then the branched transport cost of $\F$ is given by

\begin{equation*}
\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]=\int_S\tau (|\xi|)\,\mathrm{d}\Ha+\tau'(0)|\F^\perp|(\mathcal{C}).
\end{equation*}

Moreover, we can always choose $S=\{ \Theta^{*1}(|\F|,.)>0\}$ and $\F^\perp=\F\mres (\mathcal{C}\setminus S)$.

The set $S=\{ \Theta^{*1}(|\F|,.)>0\}$ is Borel measurable by <cit.>.

[$S$ not closed in general]

For polyhedral mass fluxes (that are supported on finitely many line segments) the set $S$ can clearly be chosen to be closed. In general this is not the case. A simple example is given by $\tau (m)=m$, $\mu_+=\sum_{x\in\mathbb{Q}\cap [0,1]}\varphi (x)\delta_{(x,0)}$ and $\mu_-=\sum_{x\in\mathbb{Q}\cap [0,1]}\varphi (x)\delta_{(x,1)}$, where $\varphi:\mathbb{Q}\cap [0,1]\to [0,1]$ satisfies $\sum_x\varphi (x)=1$ (see <ref>).

[->] (1,0) – (1,0.3);
[->] (2,0) – (2,0.8);
[->] (2.5,0) – (2.5,0.6);
[->] (3,0) – (3,0.5);

(1,0) – (1,4);
(2,0) – (2,4);
(2.5,0) – (2.5,4);
(3,0) – (3,4);

[label=[label distance=-0.225cm]180: $\varphi(x)$] at (2,0.4) ;

[label=-90: $S=(\mathbb{Q}\cap[0,1])\times[0,1]$] at (2,-0.2) ;

[circle,fill=black,inner sep=0.5pt,minimum size=0.1cm] at (1,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.1cm,label=-90: $x$] at (2,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.1cm] at (2.5,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.1cm] at (3,0) ;

[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (0.5,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (1.5,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.25,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.75,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (3.5,0) ;

[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (0.25,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (1.25,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.125,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.625,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (3.25,0) ;

[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (0.75,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (1.75,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.375,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.875,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (3.75,0) ;

[draw=black,circle,fill=white,inner sep=0.5pt,minimum size=0.1cm] at (1,4) ;
[draw=black,circle,fill=white,inner sep=0.5pt,minimum size=0.1cm] at (2,4) ;
[draw=black,circle,fill=white,inner sep=0.5pt,minimum size=0.1cm] at (2.5,4) ;
[draw=black,circle,fill=white,inner sep=0.5pt,minimum size=0.1cm] at (3,4) ;

[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (0.5,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (1.5,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.25,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.75,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (3.5,4) ;

[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (0.25,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (1.25,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.125,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.625,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (3.25,4) ;

[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (0.75,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (1.75,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.375,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (2.875,4) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.01cm] at (3.75,4) ;

Sketch for <ref>.

Using <cit.> it is easy to see that the following properties of $\xi$ and $\F^\perp$ hold true.
[Integrability of mass density and property of diffuse part]

Assume that $\F\in\mathcal{DM}^n(\mathcal{C})$ satisfies $\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]<\infty$ and write $\F=\xi\Ha\mres S+\F^\perp$ as in <ref>. Then the function $\xi$ is integrable with respect to $\Ha\mres S$ and $\F^\perp\mres S=0$. Those properties are independent of the triple $(\xi,S,\F^\perp)$.

By <cit.> every $\F\in\mathcal{DM}^n(\mathcal{C})$ can be written as $\F=\vartheta\Ha\mres M+\G+\psi\mathcal{L}^n\mres\mathcal{C}$ with $M\subset\mathcal{C}$ countably $1$-rectifiable, $\vartheta\in L^1(\Ha\mres M;\R^n)$ tangent to $M$ $\Ha$-almost everywhere, $\G\in\mathcal{M}^n(\mathcal{C})$ $\Ha$-diffuse and $\mathcal{L}^n$-singular as well as $\psi\in L^1(\mathcal{L}^n\mres\mathcal{C};\R^n)$. If $\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]<\infty$, we have

\begin{equation*}
\F=\xi\Ha\mres S+\F^\perp=\vartheta\Ha\mres M+\G+\psi\mathcal{L}^n\mres\mathcal{C}
\end{equation*}

with $S,\xi,\F^\perp$ as in <ref>. If $n>1$, we get $(\G+\psi\mathcal{L}^n\mres\mathcal{C})\mres S=0$. Using this and $\F^\perp\perp\Ha\mres S$ (<ref>) we obtain $\F^\perp\mres S=0$ and $\xi\Ha\mres S=\vartheta\Ha\mres (M\cap S)$, which yields $\xi\in L^1(\Ha\mres S;\R^n)$. For the case $n=1$ we can write $\xi\Ha\mres S+\F^\perp=\zeta\Ha\mres\mathcal{C}$ with the $(\Ha\mres\mathcal{C})$-integrable function

\begin{equation*}
\zeta=\begin{cases*}
\vartheta+\psi&on $M$,\\
\psi&\textup{on} $\mathcal{C}\setminus M$.
\end{cases*}
\end{equation*}

The same argument as for the case $n>1$ then yields $\xi\in L^1(\Ha\mres S;\R^n)$ and $\F^\perp\mres S=0$.

For the next result we will need the notion of irrigation patterns (see <cit.>), which are an alternative way to describe mass fluxes involving a time dependency. Broadly speaking, a mass flux between $\mu_+$ and $\mu_-$ can be seen as a superposition of particle trajectories. The so-called standard space $([0,1],\mathcal{B}([0,1]),\mathcal{L}\mres [0,1])$ can be used as a parameterization of all particles <cit.>.
[Reference space, irrigation pattern between $\mu_+,\mu_-$, total mass flux through $x$]
We define a map $\chi:[0,1]\times I\to\mathcal{C}$ such that $\chi(p,t)$ describes the position of particle $p$ at time $t$.

* The reference space for particles is the measure space $([0,1],\mathcal{B}([0,1]),\mathcal{L}\mres [0,1])$.

* An irrigation pattern is a Borel measurable map $\chi:[0,1]\times I\to\mathcal{C}$ such that $\chi(p,.)$ is absolutely continuous for $\mathcal{L}$-almost all $p\in [0,1]$.

Let $\chi$ be an irrigation pattern.

* We say that $\chi$ is an irrigation pattern between the probability measures $\mu_+$ and $\mu_-$ if and only if

\begin{equation*}
\mu_+(B)=\mathcal{L}(\{ p\in [0,1]\,|\,\chi(p,0)\in B \})\qquad\textup{and}\qquad\mu_-(B)=\mathcal{L}(\{ p\in [0,1]\,|\,\chi(p,1)\in B\})
\end{equation*}

for all $B\in\mathcal{B}(\mathcal{C})$.

* For $x\in\mathcal{C}$ we set $[x]_\chi=\{ p\in [0,1]\,|\, x\in\chi(p,I) \}$. The total mass flux through $x$ is defined by $m_\chi(x)=\mathcal{L}([x]_\chi)$.

The following summarizing statement will be used in <ref> to explicitly construct a lower semi-continuous friction coefficient $b:S\to[0,\infty)$ based on $x\mapsto|\xi(x)|$. Note that the first two points follow directly from <ref>.
[Other properties of mass density]

Assume that there is a mass flux $\G\in\mathcal{DM}^n(\mathcal{C} )$ with $\mathcal{J}^{\tau,\mu_+,\mu_-}[\G]<\infty$. Then there exists some $\F\in\mathcal{DM}^n(\mathcal{C} )$ with $\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]\leq\mathcal{J}^{\tau,\mu_+,\mu_-}[\G]$ and decomposition $\F=\xi\Ha \mres S+\F^\perp $ as in <ref> such that

* $S$ is countably $1$-rectifiable and Borel measurable,

* $\xi\in L^1(\Ha\mres S;\R^n)$ and $\F^\perp\mres S=0$,

* we can choose a representative of $\xi$ such that $|\xi|$ is bounded by the total mass $\mu_+(\mathcal{C})=1$ on $S$, $S\ni x\mapsto |\xi(x)|$ is upper semi-continuous, $\{ |\xi |\geq m \}=\{ x\in S\,|\, |\xi(x)|\geq m \}$ is closed in $\mathcal{C}$ and $\Ha(\{ |\xi |\geq m \})<\infty$ for every $m>0$,

* we can write $S=\bigcup_{m>0}\{ |\xi |\geq m \}$.

By <ref> we can write $\G=\xi_\G\Ha\mres S+\G^\perp$ with $S$ countably $1$-rectifiable and Borel measurable, $\xi_\G\in L^1(\Ha\mres S;\R^n)$ and $\G^\perp\in\mathcal{M}^n(\mathcal{C})$ with $\G^\perp\mres S=0$. As in the proof of the first inequality of <ref> in <ref> we use the idea in the proof of <cit.>; we briefly recapitulate the steps: By <cit.> we have $\G=\F_0+\F$, where $\textup{div}(\F_0)=0$ and $\F$ can be decomposed into simple oriented curves of finite length, i.e., into measures of type $\F_\gamma$ from <ref> (see also <cit.>). The measure $\F$ can be decomposed as $\F=\xi\Ha \mres S+\F^\perp\in\mathcal{DM}^n(\mathcal{C})$ with $|\xi|\leq|\xi_\G|$ and $|\F^\perp|(\mathcal{C})\leq|\G^\perp|(\mathcal{C})$ and satisfies $\textup{div}(\F)=\mu_+-\mu_-$ as well as $\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]\leq\mathcal{J}^{\tau,\mu_+,\mu_-}[\G]$. Furthermore, $\F$ can be associated with a mass flux measure $\eta$ on $\Theta$ moving $\mu_+$ onto $\mu_-$ (recall <ref>) by <cit.>. More precisely, we have

\begin{align*}
\int_{\mathcal{C}}\varphi\cdot\,\mathrm{d}\F&=\int_{\Theta }\int_I\varphi (\gamma (t) )\cdot\dot{\gamma }(t)\,\mathrm{d}\Le(t)\,\mathrm{d}\eta (\gamma )\quad\textup{ for all }\varphi\in C(\mathcal{C};\R^n)\textup{ and}\\
\int_{\mathcal{C} }\psi\,\mathrm{d}|\F|&=\int_{\Theta }\int_I\psi (\gamma (t))|\dot{\gamma (t)}|\,\mathrm{d}\Le(t)\,\mathrm{d}\eta (\gamma )\quad\textup{ for all }\psi\in C(\mathcal{C}).
\end{align*}

Using Skorohod's theorem <cit.> there is an irrigation pattern $\chi_\F$ between $\mu_+$ and $\mu_-$ which induces $\eta$,

\begin{align*}
\int_{\mathcal{C}}\varphi\cdot\,\mathrm{d}\F&=\int_{[0,1] }\int_I\varphi (\chi_\F(p,t) )\cdot \dot{\chi }_\F(p,t)\,\mathrm{d}\Le(t)\,\mathrm{d}\Le (p)\quad\textup{ for all }\varphi\in C(\mathcal{C};\R^n)\textup{ and}\\
\int_{\mathcal{C} }\psi\,\mathrm{d}|\F|&=\int_{[0,1] }\int_I\psi (\chi_\F(p,t))|\dot{\chi }_\F(p,t)|\,\mathrm{d}\Le(t)\,\mathrm{d}\Le (p) \quad\textup{ for all }\psi\in C(\mathcal{C}).
\end{align*}

Here $\dot{\chi}_\F$ denotes the derivative with respect to the second argument (which exists $\Le\mres I$-almost everywhere). By <cit.> we may assume $S=\{ m_{\chi_\F}>0\}$. Additionally, $m_{\chi_\F}(x)=|\xi (x)|$ for $\Ha$-almost every $x\in S$ by the proof of <cit.>. This shows the desired formula for $S$ by changing $\xi$ such that $m_{\chi_\F}=|\xi |$ on $S$. More specifically, represent $\xi$ by

\begin{equation*}
\tilde{\xi}=\begin{cases}
\xi&\textup{on }\{ |\xi| =m_{\chi_\F}\},\\
(m_{\chi_\F},0,\ldots ,0)^T&\textup{on }\{ |\xi| \neq m_{\chi_\F}\}.
\end{cases}
\end{equation*}

Now fix $m>0$ and let $(x_i)\subset\{|\xi|\geq m\}$ be a sequence with $x_i\to x\in\mathcal{C}$. By <cit.> the function $\mathcal{C}\ni y\mapsto m_{\chi_\F}(y)$ is upper semi-continuous. This implies $m_{\chi_\F}(x)\geq\limsup_im_{\chi_\F}(x_i)=\limsup_i|\xi (x_i)|\geq m$. In particular, we have $m_{\chi_\F}(x)>0$ and thus $x\in S$, which implies $|\xi(x)|=m_{\chi_\F}(x)\geq m$. This proves the closedness of $\{|\xi|\geq m\}$. Moreover, $|\xi|$ is upper semi-continuous by $|\xi|=m_{\chi_\F}$ on $S$. The boundedness of $|\xi|$ follows from

\begin{equation*}
|\xi|=m_{\chi_\F}=\mathcal{L}([.]_{\chi_\F})\leq 1.
\end{equation*}

Finally, we have

\begin{align*}
\infty>\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]=\int_S\tau (|\xi |)\,\mathrm{d}\Ha+\tau'(0)|\F^\perp |(\mathcal{C} )\geq\int_{\{ |\xi|\geq m\}}\tau (|\xi |)\,\mathrm{d}\Ha\geq \int_{\{ |\xi|\geq m\}}\tau (m)\,\mathrm{d}\Ha=\tau (m)\Ha(\{ |\xi|\geq m\})
\end{align*}

and thus $\Ha(\{ |\xi|\geq m\})<\infty$ for all $m>0$.

We end this subsection by reformulating the branched transport problem such that the variables to be optimized are as in the setting of the Beckmann problem from <ref>.

The branched transport problem can be written as

\begin{equation*}
\inf_{\F\in\mathcal{DM}^n(\mathcal{C}) }\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]=\inf_{S,\xi,\F^\perp}\int_S\tau (|\xi|)\,\mathrm{d}\Ha+\tau'(0)|\F^\perp|(\mathcal{C})
\end{equation*}

with $S\subset \mathcal{C}$ countably $1$-rectifiable and Borel measurable, $\xi\in L^1(\Ha\mres S;\R^n)$ and $\F^\perp\in\mathcal{M}^n(\mathcal{C})$ with $\F^\perp\mres S=0$ and $\textup{div}(\xi\Ha\mres S+\F^\perp)=\mu_+-\mu_-$.

By <ref> the right-hand side is automatically smaller than or equal to the left-hand side. For the reverse inequality, let $S,\xi,\F^\perp$ satisfy the stated properties. We can assume that $\textup{div}(\xi\Ha\mres S+\F^\perp)=\mu_+-\mu_-$ (otherwise the inequality is obvious). By <cit.> we have $\xi\Ha\mres S+\F^\perp=\vartheta\Ha\mres M+\G$ with $M$ countably $1$-rectifiable, $\vartheta\in L^1(\Ha\mres M;\R^n)$ tangent to $M$ $\Ha\mres M$-almost everywhere and $\G$ $\Ha$-diffuse. This is an admissible decomposition in the sense of <ref>. We have $\xi\Ha\mres S=\vartheta\Ha\mres (M\cap S)$ and $\xi=\vartheta$ $\Ha$-almost everywhere on $M\cap S$ as well as $\xi=0$ $\Ha$-almost everywhere on $S\setminus M$. Additionally, we get $\F^\perp=\F^\perp\mres (\mathcal{C}\setminus S)=\vartheta\Ha\mres (M\setminus S)+\G\mres (\mathcal{C}\setminus S)$. To conclude, we estimate

\begin{align*}
\mathcal{J}^{\tau,\mu_+,\mu_-}[\vartheta\Ha\mres M+\G]&=\int_M\tau (|\vartheta|)\,\mathrm{d}\Ha+\tau'(0)|\G|(\mathcal{C})=\int_{M\cap S}\tau (|\xi|)\,\mathrm{d}\Ha+\int_{M\setminus S}\tau (|\vartheta|)\,\mathrm{d}\Ha+\tau'(0)|\G|(\mathcal{C})\\
&\leq \int_{M\cap S}\tau (|\xi|)\,\mathrm{d}\Ha+\tau'(0)\int_{M\setminus S} |\vartheta|\,\mathrm{d}\Ha+\tau'(0)|\G|(\mathcal{C})\\
&= \int_{S}\tau (|\xi|)\,\mathrm{d}\Ha+\tau'(0)|\vartheta\Ha\mres (M\setminus S)+\G|(\mathcal{C})\\
&=\int_{S}\tau (|\xi|)\,\mathrm{d}\Ha+\tau'(0)|\F^\perp|(\mathcal{C}).\qedhere
\end{align*}

§.§ Duality for the total network transportation cost

We apply a generalization of Rockafellar's duality theorem between a pair of decomposable spaces of vector-valued functions <cit.>. Let $S\subset\mathcal{C}$ be countably $k$-rectifiable and $\mathcal{H}^k$-measurable. We denote the $\sigma$-algebra of $\mathcal{H}^k$-measurable subsets of $S$ by $\mathscr{H}^k(S)$. Using that the restriction of an outer measure on $\R^n$ to the Carathéodory measurable sets yields a complete measure space this property is adopted by $(S,\mathscr{H}^k(S),\mathcal{H}^k)$. Furthermore, the measure space $(S,\mathscr{H}^k(S),\mathcal{H}^k)$ is $\sigma$-finite (every member of an approximating sequence for $S$ has finite measure).

In this section “measurability” and “integrability” will refer to the measure space $(S,\mathscr{H}^k(S),\mathcal{H}^k)$.

In our case, the “vector-valued functions” are given through integrable and real-valued functions on $S$. We consider the following vector spaces:

\begin{align*}
\mathscr{L}&=\{ f:S\to\R\textup{ measurable and integrable} \} ,\\
\mathscr{L}_{\textup{b}}&=\{ f\in\mathscr{L}\, |\, f\textup{ bounded}\},\\
\mathscr{M}_{\textup{b}}&=\{ f:S\to\R\textup{ measurable and bounded}\}.
\end{align*}

[Decomposable <cit.>]

Let $V\subset\mathscr{L}$ be a subspace. $V$ is said to be decomposable if $1_Af+1_{S\setminus A}g\in V$ for all $A\in\mathscr{H}^k(S)$ with $\mathcal{H}^k(A)<\infty$, $f\in\mathscr{M}_{\textup{b}}$ and $g\in V$.

The vector spaces $\mathscr{L}$ and $\mathscr{L}_{\textup{b}}$ are decomposable: Let either $V=\mathscr{L}$ or $V=\mathscr{L}_{\textup{b}}$ and $A,f,g$ as in <ref>. Then we have

\begin{equation*}
\int_S|1_Af+1_{S\setminus A}g|\,\mathrm{d}\mathcal{H}^k\leq\mathcal{H}^k(A)\sup_A|f|+\int_{S\setminus A}|g|\,\mathrm{d}\mathcal{H}^k<\infty
\end{equation*}

and thus $1_Af+1_{S\setminus A}g\in\mathscr{L}$. Further, if $V=\mathscr{L}_{\textup{b}}$, then we obtain $|1_Af+1_{S\setminus A}g|\leq |f|+|g|\leq const.<\infty$. Moreover, we get $x\mapsto f(x)g(x)\in\mathscr{L}$ for all $f\in\mathscr{L},g\in \mathscr{L}_{\textup{b}}$ (cf. <cit.>), which needs to be satisfied to apply <cit.>.

For $g\in\mathscr{L}$ and lower semi-continuous functions $f:\R\to (-\infty,\infty ]$ we write

\begin{equation*}
\int_Sf(g)\,\mathrm{d}\mathcal{H}^k& \textup{if }\int_Sf(g)^+\,\mathrm{d}\mathcal{H}^k<\infty,\\
\infty&\textup{else},
\end{cases}
\end{equation*}

where $f(g)^+$ denotes the positive part of $f(g):S\to  (-\infty,\infty ]$. By our construction we can apply the following statement to generate the variables $b:S\to[0,\infty)$ of the urban planning problem.

Let $f:\R\to (-\infty,\infty ]$ be lower semi-continuous. Assume that $I_f(g_0)$ is finite for at least one $g_0\in\mathscr{L}_{\textup{b}}$. Then we obtain

\begin{equation*}
\end{equation*}

for all $h\in\mathscr{L}$.

Recall that $f^*$ is convex and lower semi-continuous <cit.>.

Fix a maintenance cost $\varepsilon$ induced by a transportation cost $\tau$, i.e., $\varepsilon(b)=(-\tau)^*(-b)$ ($\tau(m)=-\infty$ if $m<0$). Then the following formula will be used to replace the network-cost term in <ref>.
[Substitution of maintenance cost]

Assume that $S\subset \mathcal{C}$ is countably $k$-rectifiable and measurable with $\mathcal{H}^k(S)<\infty$. Furthermore, let $\xi :S\to\R^n$ be measurable with $\int_S|\xi|\,\mathrm{d}\mathcal{H}^k<\infty$. Assume that $\tau $ is right-continuous in $0$. Then we have

\begin{equation*}
\int_S\tau (|\xi |)\,\mathrm{d}\mathcal{H}^k=\inf_b\int_Sb|\xi |\,\mathrm{d}\mathcal{H}^k+\int_S\varepsilon (b)\,\mathrm{d}\mathcal{H}^k,
\end{equation*}

where the infimum is taken over $b\in\mathscr{L}_{\textup{b}}$ with $c=\inf\textup{dom}(\varepsilon)\leq b\leq\tau'(0)$.

The biconjugate equals the convex and lower semi-continuous envelope <cit.>. The function $-\tau$ is already convex and lower semi-continuous by the assumption that it is right-continuous in $0$. Thus, we conclude

\begin{equation*}
\tau (m)=-(-\tau )^{**}(m)=-\left(\sup_{v\in\R}vm-(-\tau )^*(v)\right)=-\left(\sup_{v\in\R}-vm-\varepsilon (v)\right)=-\varepsilon^*(-m).
\end{equation*}

Furthermore, we have $I_{\varepsilon}(v)<\infty$ for all $v\in\R$ with $\varepsilon (v)<\infty$ using $\mathcal{H}^k(S)<\infty$. We presumed $|\xi|\in\mathscr{L}$ and can therefore apply <ref>:

\begin{align*}
\int_S\tau (|\xi|)\,\mathrm{d}\mathcal{H}^k=-\int_S\varepsilon^*(-|\xi|)\,\mathrm{d}\mathcal{H}^k=-I_{\varepsilon^*}(-|\xi|)=-\sup_{b\in\mathscr{L}_{\textup{b}}}-\int_Sb|\xi|\,\mathrm{d}\mathcal{H}^k-I_\varepsilon (b)=\inf_{b\in\mathscr{L}_{\textup{b}}}\int_Sb|\xi|\,\mathrm{d}\mathcal{H}^k+\int_S\varepsilon (b)\,\mathrm{d}\mathcal{H}^k.
\end{align*}

We can require $b\geq c$, because $\varepsilon (v)=\infty$ for $v<c$. Moreover, if $b\in\mathscr{L}_{\textup{b}}$ with $b\geq c$ such that the expression on the right-hand side is finite, then $\tilde{b}=\min\{ b,\tau'(0) \}$ satisfies $\tilde{b}\leq\tau'(0)$ and

\begin{equation*}
\int_S\tilde{b}|\xi|\,\mathrm{d}\mathcal{H}^k+\int_S\varepsilon (\tilde{b})\,\mathrm{d}\mathcal{H}^k\leq \int_Sb|\xi|\,\mathrm{d}\mathcal{H}^k+\int_S\varepsilon (\tilde{b})\,\mathrm{d}\mathcal{H}^k=\int_Sb|\xi|\,\mathrm{d}\mathcal{H}^k+\int_S\varepsilon (b)\,\mathrm{d}\mathcal{H}^k
\end{equation*}

by using $\varepsilon(v)=(-\tau )^*(-v)=0$ for $v\geq \tau'(0)$.

The condition $\mathcal{H}^k(S)<\infty$ is necessary for the case $\tau'(0)=\infty$: Assume that $b\in\mathscr{L}_{\textup{b}}$ and $\mathcal{H}^k(S)=\infty$, then

\begin{equation*}
\int_S\varepsilon(b)\,\mathrm{d}\mathcal{H}^k\geq \varepsilon (\sup b)\mathcal{H}^k(S)=\infty
\end{equation*}

if $\tau'(0)=\infty$, but $\int_S\tau (|\xi|)\,\mathrm{d}\mathcal{H}^k$ can be finite for appropriate $\xi\in L^1(\mathcal{H}^k\mres S;\R^n)$. Nevertheless, we get the desired result for $\mathcal{H}^k(S)=\infty$ by relaxing the properties of $b$.

Let $S\subset\mathcal{C}$ be countably $k$-rectifiable and measurable. Furthermore, let $\xi :S\to\R^n$ be measurable with $\int_S|\xi|\,\mathrm{d}\mathcal{H}^k<\infty$. Assume that $\tau $ is right-continuous in $0$. Then we have

\begin{equation*}
\int_S\tau (|\xi |)\,\mathrm{d}\mathcal{H}^k=\inf_b\int_Sb|\xi |\,\mathrm{d}\mathcal{H}^k+\int_S\varepsilon (b)\,\mathrm{d}\mathcal{H}^k,
\end{equation*}

where the infimum is taken over measurable functions $b: S\to [0,\infty )$ with $\inf\textup{dom}(\varepsilon)\leq b\leq\tau'(0)$.

Using the procedure in <ref> we can assume that $S$ is a disjoint countable union of finitely $k$-rectifiable sets $S_i$, i.e., $S_i\subset S$ are measurable with $\mathcal{H}^k(S_i)<\infty$ (we neglect zero sets due to the integration). Let $\delta>0$ and $\delta_i>0$ such that $\sum_i\delta_i=\delta$. Assume that $b_i$ are as in <ref> with

\begin{equation*}
\int_{S_i}\tau (|\xi |)\,\mathrm{d}\mathcal{H}^k\geq\int_{S_i}|\xi|b_i+\varepsilon (b_i)\,\mathrm{d}\mathcal{H}^k-\delta_i.
\end{equation*}

Define $\tilde{b}:S\to\R$ by $\tilde{b}=b_i$ on $S_i$. Then $\tilde{b}$ has the desired properties, i.e., $\tilde{b}$ is measurable with $\inf\textup{dom}(\varepsilon)\leq \tilde{b}\leq\tau'(0)$. Application of the monotone convergence theorem yields

\begin{align*}
\int_S\tau (|\xi|)\,\mathrm{d}\mathcal{H}^k&= \sum_i\int_{S_i}\tau (|\xi|)\,\mathrm{d}\mathcal{H}^k
\geq \sum_i\int_{S_i}|\xi|b_i+\varepsilon (b_i)\,\mathrm{d}\mathcal{H}^k-\delta
=\int_{S}|\xi|\tilde{b}+\varepsilon (\tilde{b})\,\mathrm{d}\mathcal{H}^k-\delta\\
&\geq\inf_b\int_{S}|\xi|b+\varepsilon (b)\,\mathrm{d}\mathcal{H}^k-\delta
=\inf_b\int_{S}|\xi|b\,\mathrm{d}\mathcal{H}^k+\int_S\varepsilon (b)\,\mathrm{d}\mathcal{H}^k-\delta,
\end{align*}

where the infima are taken over functions $b$ like in the statement. Notice that $\varepsilon,b\geq 0$ was used in the last equation. The reverse inequality is a direct consequence of the definition of the convex conjugate. More precisely, let $b:S\to [0,\infty)$ be any measurable function with $\inf\textup{dom}(\varepsilon)\leq b\leq\tau'(0)$. Then we obtain

\begin{equation*}
\int_S\tau (|\xi|)\,\mathrm{d}\mathcal{H}^k=-\int_S\varepsilon^*(-|\xi|)\,\mathrm{d}\mathcal{H}^k=-\int_S\sup_{v\in\R}-v|\xi (x)|-\varepsilon (v)\,\mathrm{d}\mathcal{H}^k(x)\leq\int_Sb|\xi|\,\mathrm{d}\mathcal{H}^k+\int_S\varepsilon(b)\,\mathrm{d}\mathcal{H}^k
\end{equation*}

by using $v=v(x)=b(x)$ in the inequality. Taking the infimum over $b$ shows the stated formula.

In the setting of this subsection we can assume that the functions $b$ in <ref> are lower semi-continuous.

The functions $b$ in <ref> can be assumed to be lower semi-continuous if $\tau'(0)<\infty$.

As a first step we simplify the the statement to be shown. Write $c=\inf\textup{dom}(\varepsilon)$. For $f\in\mathscr{M}_{\textup{b}}$ with $f\geq 0$ define

\begin{equation*}
F(f)=\int_Sf|\xi|+\varepsilon (f)\,\mathrm{d}\mathcal{H}^k\in [0,\infty ].
\end{equation*}

Observe that there exists a function $b\in\mathscr{M}_{\textup{b}}$ with $F(b)<\infty$ (e.g., $b\equiv\tau'(0)$). Let $S^N$ be an approximating sequence for $S$. Define functions belonging to $S^N$ by

\begin{equation*}
b&\textup{on }S^N,\\
\tau'(0)&\textup{on }S\setminus S^N.
\end{cases}
\end{equation*}

We have $\varepsilon (b_N)\nearrow\varepsilon (b)$ and $b_N|\xi|\searrow b|\xi|$ for $N\to\infty$ pointwise $\mathcal{H}^k\mres S$-almost everywhere. Furthermore, the $b_N|\xi|$ satisfy

\begin{equation*}
\int_Sb_N|\xi|\,\mathrm{d}\mathcal{H}^k\leq\tau'(0)\int_S|\xi|\,\mathrm{d}\mathcal{H}^k<\infty.
\end{equation*}

Hence, $F(b_N)\to F(b)$ for $N\to\infty$ by the monotone convergence theorem. Thus, for fixed $N$ it is enough to find a lower semi-continuous funcion $l_N:S\to [c,\tau'(0)]$ such that $F(l_N)$ is arbitrarily close to $F(b_N)$. This reduces to the problem of finding a lower semi-continuous function $l_N:S^N\to [c,\tau'(0)]$ with $F_N(l_N)$ being arbitrarily close to $F_N(b_N)$, where

\begin{equation*}
F_N(f)=\int_{S^N}f|\xi|+\varepsilon (f)\,\mathrm{d}\mathcal{H}^k
\end{equation*}

for $f\in\mathscr{M}_{\textup{b}}$ with $f\geq 0$, because we can then replace $l_N$ by the lower semi-continuous function ($S^N$ is closed)

\begin{equation*}
\tilde{l}_N=\begin{cases}
l_N&\textup{on }S^N,\\
\tau'(0)&\textup{on }S\setminus S^N.
\end{cases}
\end{equation*}

For simplicity, we write $S,F,b$ instead of $S^N,F_N,b_N$ and assume that $S$ is closed with $\mathcal{H}^k(S)<\infty$ (like the $S^N$) for the rest of the proof. Now the first task is to show that $b$ can be assumed to be Borel measurable on $S$ ($b$ is measurable with respect to $\mathcal{H}^k\mres S$). Write $b_\delta=b+\delta$ for $\delta>0$. By <ref> we can assume $b_\delta\in I_\delta= [c+\delta,\tau'(0)+\delta ]$. For $K\in\mathbb{N}$ divide $I_\delta$ into disjoint intervals $I_1^K,\ldots ,I_K^K$ such that $|I_i^K|=(\tau'(0)-c)/K$ for $i=1,\ldots ,K$ and $t_1<t_2$ if $t_1\in I_i^K,t_2\in I_j^K$ and $i<j$. By <cit.> we can write $b_\delta^{-1}(I_i^K)=A_i^K\dot{\cup }B_i^K$ with $B_i^K$ Borel measurable and $\mathcal{H}^k(A_i^K)<1/K^2$ (by assumption $\mathcal{H}^k(b_\delta^{-1}(I_i^K))\leq\mathcal{H}^k(S)<\infty$). For $x\in S$ define the functions

\begin{equation*}
c+\delta&\textup{if }x\in\bigcup_{i=1}^KA_i^K,\\
\inf I_i^K&\textup{if }x\in B_i^K\textup{ for some } i\in\{ 1,\ldots ,K\}.
\end{cases}
\end{equation*}

For all $K\in\mathbb{N}$ and $t\geq c+\delta$ we have

\begin{equation*}
\{ g_\delta^K\leq t\} =\left(\bigcup_{i=1}^KA_i^K\right)\cup\bigcup_{\inf I_i^K\leq t}B_i^K=\left(S\bigg\backslash\bigcup_{i=1}^KB_i^K\right)\cup\bigcup_{\inf I_i^K\leq t}B_i^K.
\end{equation*}

This shows that $g_\delta^K$ is Borel measurable. Moreover, we get

\begin{align*}
\int_S|g_\delta^K-b_\delta|\,\mathrm{d}\mathcal{H}^k&=\sum_{i=1}^{K}\left(\int_{A_i^K}|g_\delta^K-b_\delta|\,\mathrm{d}\mathcal{H}^k+\int_{B_i^K}|g_\delta^K-b_\delta|\,\mathrm{d}\mathcal{H}^k\right)
\leq\sum_{i=1}^{K}(\tau'(0)-c)\mathcal{H}^k(A_i^K)+\frac{\tau'(0)-c}{K}\mathcal{H}^k(S)\\
&\leq \frac{1}{K}(\tau'(0)-c)(1+\mathcal{H}^k(S))
\end{align*}

which tends to $0$ for $K\to\infty$. This yields the existence of a subsequence $g_\delta^{K_i}$ with $g_\delta^{K_i}\to b_\delta$ pointwise $(\mathcal{H}^k\mres S)$-almost everywhere for $i\to\infty$. Now let $f_\delta^{K_i}=g_\delta^{K_i}|\xi|+\varepsilon (g_\delta^{K_i})$. We obtain $|f_\delta^{K_i}|\leq (\tau'(0)+\delta)|\xi|+\varepsilon (c+\delta)\in\mathscr{L}=\mathscr{L}(S)$ and $f_\delta^{K_i}\to b_\delta|\xi|+\varepsilon (b_\delta )$ pointwise $\mathcal{H}^k\mres S$-almost everywhere for $i\to\infty$ ($\varepsilon$ is continuous on $(c,\infty )$). Thus, we have $F(g_\delta^{K_i})\to F(b_\delta)$ for $i\to\infty$ by Lebesgue's dominated convergence theorem. In addition, we have $h_\delta=b_\delta|\xi|+\varepsilon (b_\delta)\to b|\xi|+\varepsilon (b)$ for $\delta\to 0$ ($\varepsilon$ is right-continuous in $c$) and $|h_\delta|\leq (\tau'(0)+\delta)|\xi|+\varepsilon (c)\in\mathscr{L}$ which yields $F(b_\delta )\to F(b)$ for $\delta\to 0$. This shows that $b$ can be assumed to be Borel measurable by choosing $\delta$ sufficiently small and then $i$ sufficiently large (for the corresponding sequence $g_\delta^{K_i}$). For the approximation with lower semi-continuous functions we need some regularity properties of $\mathcal{H}^k\mres S$. Let $B\in\mathcal{B}(S)$. By <cit.> there exists a countable intersection of open sets $U=\bigcap_iU_i$ such that

\begin{equation*}
\mathcal{H}^k(B)=\mathcal{H}^k(S\cap U)=\lim_{M\to\infty }\mathcal{H}^k\left( S\cap\bigcap_{i=1}^MU_i\right).
\end{equation*}

By the fact that $S\cap\bigcap_{i=1}^MU_i$ is open in the subspace topology we get that $\mathcal{H}^k$ is outer regular on $S$. Additionally, $\mathcal{H}^k$ is tight on $S$ <cit.>. Hence, we can apply the Vitali–Carathéodory theorem <cit.> which yields: for all $\delta>0$ there exists a sequence of lower semi-continuous functions $l_i$ with $l_i\leq b_\delta$ on $S$, $l_i\geq c+\delta/2$ and

\begin{equation*}
\int_S(b_\delta-l_i)\,\mathrm{d}\mathcal{H}^k\to 0\textup{ for }i\to\infty.
\end{equation*}

Likewise, the same argumentation as above yields the desired result.

§.§ Branched transport problem as generalized urban planning problem

In this section we finally prove <ref>, essentially by constructing a minimizer for one problem from one of the other. We will also discuss a few examples illustrating the relation between the minimizers. Throughout we assume that $(\tau,\mu_+,\mu_-)$ is a triple for the branched transport problem and the corresponding maintenance cost $\varepsilon$ is defined via $\tau$ as in <ref>.

Let $\tau_+(0)=\lim_{m\searrow 0}\tau(m)$. We define

\begin{equation*}
\tilde{\tau}(m)=\begin{cases*}
\tau(m)&if $m\leq 0$,\\
\tau(m)-\tau_+(0)&else
\end{cases*}
\quad\text{for }m\in\R
\qquad\text{and}\qquad
\tilde{\varepsilon}(v)=(-\tilde{\tau})^*(-v)
\quad\text{for }v\in\R.
\end{equation*}

We will need the following properties of $\tilde{\tau}$.

The transportation cost $\tilde{\tau}$ is right-continuous in $0$. Furthermore, we have $\tilde{\varepsilon}=\varepsilon-\tau_+(0)$ and

\begin{equation*}
\int_S\tau(|\xi|)\dHa=\int_S\tilde{\tau}(|\xi|)\dHa+\tau_+(0)\Ha(\{ |\xi|>0 \})
\end{equation*}

for all $\xi\in L^1(\Ha\mres S;\R^n)$.

The first property follows by definition. Let $v\in\R$ and assume that $\tilde{\varepsilon}(v)>0$. We have

\begin{equation*}
\tilde{\varepsilon}(v)=\sup_{m>0}\tilde{\tau}(m)-mv=\sup_{m>0}\tau(m)-\tau_+(0)-mv=\sup_{m\in\R}\tau(m)-mv-\tau_+(0)=\varepsilon(v)-\tau_+(0).
\end{equation*}

If $\tilde{\varepsilon}(v)=0$, then we get $v\geq\tilde{\tau}'(0)$ and therefore $\varepsilon(v)=\tau_+(0)$. Finally, we observe that

\begin{equation*}
\int_S\tau(|\xi|)\dHa=\int_{\{ |\xi|>0 \}}\tau(|\xi|)\dHa=\int_{\{ |\xi|>0 \}}\tilde{\tau}(|\xi|)+\tau_+(0)\dHa=\int_{\{ |\xi|>0 \}}\tilde{\tau}(|\xi|)\dHa+\tau_+(0)\Ha(\{ |\xi|>0 \}).\qedhere
\end{equation*}

For the inequality $\inf\mathcal{J}^{\tau,\mu_+,\mu_-}\geq\inf\mathcal{U}^{\varepsilon,\mu_+,\mu_-}$ we will use the following standard composition property which we state without proof.

Let $(X,d_X)$ be a metric space and $A\subset\R$. If $g:X\to A$ is upper semi-continuous and $f:A\to\R$ lower semi-continuous and decreasing, then $f\circ g$ is lower semi-continuous.

Let $\delta>0$ and $x,x_j\in X$ such that $d_X(x,x_j)\to 0$. We have $g(x_j)<g(x)+\delta$ for $j$ sufficiently large, because $g$ is upper semi-continuous. Hence, we get $f(g(x)+\delta)\leq f(g(x_j))$ for all such $j$ by the assumption that $f$ is decreasing. Thus, we have

\begin{equation*}
f(g(x)+\delta)\leq\liminf_j f(g(x_j))
\end{equation*}

for all $\delta>0$ and therefore, using that $f$ is lower semi-continuous,

\begin{equation*}
f(g(x))\leq\liminf_{\delta\to 0}f(g(x)+\delta)\leq\liminf_j f(g(x_j)).\qedhere
\end{equation*}

$\inf\mathcal{J}^{\tau,\mu_+,\mu_-}\leq\inf\mathcal{U}^{\varepsilon,\mu_+,\mu_-}$: Assume that

\begin{equation*}
\mathcal{U}^{\varepsilon,\mu_+,\mu_-}[S,b]<\infty
\end{equation*}

for some admissible pair $(S,b)$.
As discussed in the introduction of <ref>, this automatically implies the validity of <ref>.
Thus by <ref> we have

\begin{equation*}
\end{equation*}

for some $\xi\in L^1(\Ha\mres S;\R^n)$ and $\F^\perp\in\mathcal{M}^n(\mathcal{C})$ with $\F^\perp\mres S=0$.
We show that $\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]\leq\mathcal{U}^{\varepsilon,\mu_+,\mu_-}[S,b]$ for $\F=\xi\Ha\mres S+\F^\perp$. If $\tau$ is right-continuous in $0$, then $-\tau$ is lower semi-continuous and convex and thus equals its biconjugate <cit.>. This yields

\begin{equation*}
\tau (m)=-(-\tau )^{**}(m)=-\left(\sup_{v\in\R}vm-(-\tau )^*(v)\right)=-\left(\sup_{v\in\R}-vm-\varepsilon (v)\right)=-\varepsilon^*(-m)
\end{equation*}

and thus

\begin{multline*}
\mathcal{U}^{\varepsilon,\mu_+,\mu_-}[S,b]=\int_Sb|\xi|\dHa+\tau'(0)|\F^\perp|(\mathcal{C})+\int_S\varepsilon(b)\dHa\geq\int_S\inf_{v\in\R}|\xi|v+\varepsilon(v)\,\mathrm{d}\Ha+\tau'(0)|\F^\perp|(\mathcal{C})\\
\end{multline*}

by <ref>.
If $\tau$ is not right-continuous, then we have $\tau'(0)=\infty$ and therefore $\F^\perp=0$. Moreover, using <ref> and the previous estimate we obtain

\begin{multline*}
\mathcal{U}^{\varepsilon,\mu_+,\mu_-}[S,b]
\geq\int_S\tilde{\tau}(|\xi|)\dHa+\tau_+(0)\Ha(S)\\
\geq\int_S\tilde{\tau}(|\xi|)\dHa+\tau_+(0)\Ha(\{ |\xi|>0 \})
\end{multline*}

$\inf\mathcal{J}^{\tau,\mu_+,\mu_-}\geq\inf\mathcal{U}^{\varepsilon,\mu_+,\mu_-}$: Assume that there exists some $\G\in\mathcal{DM}^n(\mathcal{C})$ with $\mathcal{J}^{\tau,\mu_+,\mu_-}[\G]<\infty$. Let $\F=\xi\Ha\mres S+\F^\perp$ be as in <ref> ($\F$ can be constructed by removing divergence-free parts of $\G$). The function $S\ni x\mapsto g(x)=|\xi(x)|$ is upper semi-continuous by <ref>. Thus, $\{ |\xi|=0 \}\subset S$ is Borel measurable and we can assume without loss of generality that $S=\{ |\xi|>0 \}$. For $m>0$ we define

\begin{equation*}
\end{equation*}

which is well-defined, because the subdifferential is closed. Furthermore, $f$ is decreasing and lower semi-continuous on $(0,\infty)$ by construction. Using <ref> the function $b:S\to[0,\infty)$ defined by

\begin{equation*}
\end{equation*}

is lower semi-continuous on $S$. Additionally, we have $\varepsilon(b)=\tau(|\xi|)-|\xi|b$ by definition. Therefore, by <ref> we get

\begin{equation*}
\infty>\mathcal{J}^{\tau,\mu_+,\mu_-}[\G]\geq\mathcal{J}^{\tau,\mu_+,\mu_-}[\F]=\int_S\tau(|\xi|)\dHa+\tau'(0)|\F^\perp|(\mathcal{C})=\int_Sb|\xi|\dHa+\tau'(0)|\F^\perp|(\mathcal{C})+\int_S\varepsilon(b)\dHa.
\end{equation*}

Thus we must have $\int_S\varepsilon(b)\dHa<\infty$ which shows that <ref> is satisfied. Hence we can apply <ref> and continue the estimation,

\begin{equation*}
\mathcal{J}^{\tau,\mu_+,\mu_-}[\G]\geq\int_Sb|\xi|\dHa+\tau'(0)|\F^\perp|(\mathcal{C})+\int_S\varepsilon(b)\dHa\geq W_{d_{S,\tau'(0),b}}(\mu_+,\mu_-)+\int_S\varepsilon(b)\dHa=\mathcal{U}^{\varepsilon,\mu_+,\mu_-}[S,b].\qedhere
\end{equation*}

In the remainder of the section we discuss a few consequences of the proof.
In <cit.> it is shown that the generalized branched transport problem either has a minimizer or is infeasible,
i.e., there is no mass flux of finite energy transporting $\mu_+$ to $\mu_-$.
(Note that under additional growth conditions on $\tau$ near $0$ one can always obtain existence of a minimizer independent of $\mu_+$ and $\mu_-$ <cit.>.)
Since in the proof of <ref> we constructed from each feasible candidate for one problem a feasible candidate for the other, this immediately implies the following.
[Existence of optimizers]
The generalized branched transport problem and the associated generalized urban planning problem either both admit a minimizer or are both infeasible.

While for $\tau'(0)=\infty$ the optimal mass flux of the generalized branched transport problem is known to be rectifiable <cit.>,
one gets an even stronger result if $\tau$ is not right-continuous in $0$.

If $\tau$ is not right-continuous in $0$, then any street network $(S,b)$ with finite urban planning cost satisfies $\Ha(S)<\infty$.
Indeed, from <ref> of $\varepsilon$ we obtain $\varepsilon\geq\tau_+(0)$ so that

\begin{equation*}
\infty
\geq\int_S\varepsilon(b)\,\mathrm{d}\Ha
\geq\tau_+(0)\Ha(S).
\end{equation*}

Finally, let us briefly discuss and illustrate the relation between minimizers of the branched transport and the urban planning problem.
A natural question is whether they are in one-to-one correspondence.
However, this is not to be expected for the following reason.
In our equivalence proof, the central step to switch between the mass flux and the friction coefficient as variables was the relation

\begin{equation*}
\tau(|\xi|)=b|\xi|+\varepsilon(b),
\end{equation*}

which we exploited to construct one variable from the other.
Note that $\tau(|\xi|)\leq b|\xi|+\varepsilon(b)$ is nothing else than the Fenchel–Young inequality,
and if both $\xi$ and $b$ should be optimal one needs to have equality.
However, this equality only yields a one-to-one relation between $|\xi|$ and $b$ if both $\tau$ and $\varepsilon$ are differentiable or equivalently strictly convex.
If, however, $\tau$ has a kink at $|\xi|$, then there exist multiple $b$ satisfying the equality.
Likewise, if $\varepsilon$ has a kink at $b$, then there exist multiple solutions $|\xi|$.
Consequently, a single minimizer of the branched transport problem will sometimes correspond to multiple minimizers of the urban planning problem and vice versa.
We close this section by illustrating this fact with three examples.
The first example is standard in classical optimal transport theory and illustrates
that there may be multiple optimal mass fluxes, while the optimal friction coefficient is unique.
In fact, the classical Wasserstein cost is a rather degenerate case of branched transport for which $\inf\{ \varepsilon<\infty \}=\tau'(0)$ (see <ref>)
so that the friction coefficient (which has to lie in between both values) is uniquely fixed a priori.
We provide a less degenerate example directly after, in which the uniqueness of the friction coefficient comes from a kink in $\varepsilon$.
[Infinitely many optimal mass fluxes, but unique friction coefficient]

Let $\tau(m)=m$ and consider

\begin{equation*}
\mu_+=\tfrac{1}{2}\delta_{x_1}+\tfrac{1}{2}\delta_{x_2}\textup{\qquad and\qquad}\mu_-=\tfrac{1}{2}\delta_{y_1}+\tfrac{1}{2}\delta_{y_2}
\end{equation*}

for $x_1=(0,0)$, $x_2=(2,0)$, $y_1=(1,1)$, $y_2=(1,-1)$ (see <ref>). Then the optimal solution for the urban planning problem is given by $b=1$ $\Ha$-almost everywhere ($S$ arbitrary). Since this choice of $b$ is the only feasible one, it is unique.
Nevertheless, there exist infinitely many solutions to the branched transport problem. Each one is associated with an optimal transport plan for the Wasserstein-1-distance between $\mu_+$ and $\mu_-$, where the family of optimal transport plans can be parameterized by

\begin{equation*}
m=\pi(\{  x_1\}\times\{ y_1 \})\in[0,\tfrac12].
\end{equation*}

[thick] (0,0) – (1,1) – (2,0) – (1,-1) – (0,0);
[-Straight Barb[width=3mm,length=3mm],thick] (0,0) – (0.5,0.5);
[-Straight Barb[width=3mm,length=3mm],thick] (0,0) – (0.5,-0.5);
[-Straight Barb[width=3mm,length=3mm],thick] (2,0) – ++(-0.5,0.5);
[-Straight Barb[width=3mm,length=3mm],thick] (2,0) – ++(-0.5,-0.5);
[circle,fill=black,inner sep=0.5pt,minimum size=0.1cm] at (0,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.1cm] at (2,0) ;
[draw=black,circle,fill=white,inner sep=0.5pt,minimum size=0.1cm] at (1,1) ;
[draw=black,circle,fill=white,inner sep=0.5pt,minimum size=0.1cm] at (1,-1) ;
[label=[label distance=-0.3cm]90: $\F_m$] at (1,0) ;
[label=180: $m$] at (0.5/sqrt(2),0.5/sqrt(2)) ;
[label=0: $1/2-m$] at (2-0.5/sqrt(2),0.5/sqrt(2)) ;
[label=180: $1/2-m$] at (0.5/sqrt(2),-0.5/sqrt(2)) ;
[label=0: $m$] at (2-0.5/sqrt(2),-0.5/sqrt(2)) ;
[label=180: $x_1$] at (0,0) ;
[label=90: $y_1$] at (1,1) ;
[label=0: $x_2$] at (2,0) ;
[label=270: $y_2$] at (1,-1) ;

Sketch illustrating <ref>. The mass flux $\F_m$ is optimal for all $m\in[0,1/2]$.

[$\varepsilon$ has a kink]

Consider the initial and final mass

\begin{equation*}
\mu_+=\tfrac25\delta_{(0,0)}+\tfrac15\delta_{(1,-\ell)}+\tfrac25\delta_{(2,0)}\textup{\qquad and\qquad}\mu_-=\tfrac25\delta_{(0,1)}+\tfrac15\delta_{(1,1+\ell)}+\tfrac25\delta_{(2,1)}
\end{equation*}

with parameter $\ell> 0$.
If $\tau(\frac35)<\tau(\frac15)+\tau(\frac25)$ and $\ell$ is chosen sufficiently large we obtain two symmetric solutions for an optimal mass flux $\F$, no matter how $\tau$ looks like: The mass from $(1,-\ell)$ will be jointly transported with either the mass from $(0,0)$ or the mass from $(2,0)$ (see <ref> left). Moreover, one can argue that for large enough $\ell$ the mass from $(1,-\ell)$ will in fact in one case be transported through $(0,0)$ as well as $(0,1)$ and in the other case through $(2,0)$ as well as $(2,1)$:
Indeed, if the mass from $(0,0)$ and $(-1,\ell)$ would combine in a point $p_\ell\neq(0,0)$,
then the so-called momentum conservation (a local optimality condition at triple junctions <cit.>) reads

\begin{equation*}
\tau(\tfrac15+\tfrac25)\cdot(0,1)=\tau(\tfrac25)\cdot v_\ell+\tau(\tfrac15)\cdot w_\ell\textup{ with }v_\ell=\frac{p_\ell}{|p_\ell|}\textup{ and }w_\ell=\frac{p_\ell-(1,-\ell)}{|p_\ell-(1,-\ell)|}.
\end{equation*}

For $\ell\to\infty$ the angle between $v_\ell$ and $w_\ell$ would go to zero which would violate this equality.
Triple junctions near the other points can be excluded analogously. Now even though there are two solutions to the branched transport problem,
if $\tau$ is such that $\varepsilon$ has a kink in the right place or equivalently if $\tau$ is affine at least on $[\frac25,\frac35]$,
the corresponding urban planning problem has a unique solution.
For instance, define a differentiable transportation cost (see <ref> middle) by

\begin{equation*}
\tau(m)=\begin{cases*}
m&if $m\in[0,\tfrac15]$,\\
\frac25-\frac54(m-\frac35)^2&if $m\in(\frac15,\frac25]$,\\
\frac{3}{20}+\frac12m&if $m\in(\frac25,\frac35]$,\\
-\frac{11}{20}+\sqrt{m+\frac{2}{5}}&if $m>\frac35$.
\end{cases*}
\end{equation*}

The corresponding maintenance cost (<ref> right) is given by

\begin{equation*}
\varepsilon(b)=\begin{cases*}
\frac1{4b}-\frac{11}{20}+\frac25b&if $b\in(0,\frac12]$,\\
\frac1{5}b^2-\frac35b+\frac25&if $b\in(\frac12,1]$,\\
0&if $b>1$.
\end{cases*}
\end{equation*}

Note that $\varepsilon$ has a kink at $b=\frac12$ with left derivative $-\frac35$ and right derivative $-\frac25$.
The solution $(S,b)$ to the urban planning problem is uniquely determined in the following sense: Ignoring sets where $b=\tau'(0)=1$ (which one may always do without loss of generality) and identifying $b$ in the $\Ha$-almost everywhere sense, the optimal pair $(S,b)$ is uniquely given by

\begin{equation*}
S=[(0,0),(0,1)]\cup[(2,0),(2,1)]\textup{\qquad and\qquad}b\equiv\tfrac12.
\end{equation*}

[line width = 0.05cm] (0,0) – (0,1);
[line width = 0.05cm] (2,0) – (2,1);
(0,1) – ++(1/3,2/3);
[loosely dotted] (1/3,1+2/3) – (2/3,1+4/3);
(2/3,1+4/3) – (3/3,1+6/3);
(0,0) – ++(1/3,-2/3);
[loosely dotted] (1/3,-2/3) – (2/3,-4/3);
(2/3,-4/3) – (3/3,-6/3);
[->] (0,1) – ++(0.25*1/sqrt(5),0.3*2/sqrt(5));
[->] (0,0) – (0,0.85);
[->] (1,-2) – ++(-0.25*1/sqrt(5),0.25*2/sqrt(5));
[->] (2,0) – ++(0,0.6);
[circle,fill=black,inner sep=0.5pt,minimum size=0.1cm,label=180:$(0,0)$] at (0,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.1cm,label=0:$(2,0)$] at (2,0) ;
[circle,fill=black,inner sep=0.5pt,minimum size=0.1cm,label=0:$(1,-\ell)$] at (1,-2) ;
[draw=black,circle,fill=white,inner sep=0.5pt,minimum size=0.1cm,label=180:$(0,1)$] at (0,1) ;
[draw=black,circle,fill=white,inner sep=0.5pt,minimum size=0.1cm,label=0:$(2,1)$] at (2,1) ;
[draw=black,circle,fill=white,inner sep=0.5pt,minimum size=0.1cm,label=0:$(1,1+\ell)$] at (1,3) ;

$2$ possible mass fluxes

[x = 9.6em, y = 9.6em,
samples = 500,
domain = 0:1,
ymax = .8,
axis x line=bottom,
axis y line=left,
xlabel =  $m$,
ylabel =  $\tau (m)$,
x label style=at=(axis description cs:1,0),anchor=south,
y label style=at=(axis description cs:.07,0.9),rotate=270,anchor=west,
xtick = 0.2,0.4,0.6,
xticklabels = $\tfrac15$,$\tfrac25$,$\tfrac35$,
ytick = ,

[black,domain=0:0.2,line width=2pt] expression x;
[black,domain=0.2:0.4,line width=2pt] expression .4-5/4*(x-.6)^2;
[black,domain=0.4:0.6,line width=2pt] expression .15+.5*x;
[black,domain=0.6:.9,line width=2pt] expression -11/20+sqrt(x+.4);
[black,line width=0pt] expression 0;
[only marks,mark=*,mark options=scale=1, fill=black,text mark as node=true] coordinates(1/5,1/5);
[only marks,mark=*,mark options=scale=1, fill=black,text mark as node=true] coordinates(2/5,7/20);
[only marks,mark=*,mark options=scale=1, fill=black,text mark as node=true] coordinates(3/5,9/20);

$\tau$ differentiable

[x = 8em, y = 8em,
samples = 500,
domain = 0:1.2,
ymax = 0.96,
axis x line=bottom,
axis y line=left,
xlabel = $b$,
ylabel = $\varepsilon (b)$,
x label style=at=(axis description cs:1,0),anchor=south,
y label style=at=(axis description cs:.1,0.9),rotate=270,anchor=west,
xtick = .5,1,
xticklabels = $\frac12$,$1$,
ytick = ,

[black,domain=0.1:.5,line width=2pt] expression 0.25/x-11/20+.4*x;
[black,domain=0.5:1,line width=2pt] expression x^2/5-3*x/5+.4;
[black,domain=1:1.1,line width=2pt] expression 0;
[black,line width=0pt] expression 0;

$\varepsilon$ with kink at $b=\frac12$

Sketch illustrating <ref>. Due to a kink in $\varepsilon$ there are $2$ possible optimal mass fluxes for $\ell$ sufficiently large.

The final example illustrates the reverse situation: The transportation cost $\tau$ exhibits a nondifferentiability so that to one $\xi$ there may correspond multiple $b$.
As a result there will be a unique optimal mass flux for the branched transport problem, but multiple optimal solutions of the urban planning problem.
[$\tau$ has a kink]

Assume that $\mu_+=\delta_x$ and $\mu_-=\delta_y$ with $x\neq y$ (see <ref> left). Then the solution of the branched transport problem is unique (independent of $\tau$) and given by $\F_{opt}=\vec{e}\Ha\mres e$, where $e=[x,y]$ and $\vec{e}=(y-x)/|y-x|$. Further, we have $\mathcal{J}^{\tau,\mu_+,\mu_-}[\F_{opt}]=\tau(1)\Ha(e)$. Now set $S=e$ and let $\tau$ have a kink, for instance $\tau(m)=\min(m,1)$. Then for every spatially constant (and in fact even non-constant) $b\in [0,1]$ we obtain

\begin{equation*}
\mathcal{U}^{\varepsilon,\mu_+,\mu_-}[S,b]=W_{d_{e,1,b}}(\delta_x,\delta_y)+\int_e\varepsilon(b)\dHa=b\Ha(e)+\varepsilon(b)\Ha(e)=\Ha(e)=\mathcal{J}^{\tau,\mu_+,\mu_-}[\F_{opt}]
\end{equation*}

(see <ref> right) so that there exist infinitely many optimal friction coefficients $b$.
Note that in this example $\varepsilon$ is differentiable below $a=\tau'(0)$.

[label=180:$x$,circle,fill=black,inner sep=0.5pt,minimum size=0.1cm] (x) at (0,0) ;
[label=0:$y$,draw=black,circle,fill=white,inner sep=0.5pt,minimum size=0.1cm] (y) at (2,1.5) ;
[thick] (x) – (y);
[-Straight Barb[width=3mm,length=3mm],thick] (x) –++(1,0.75);

optimal mass flux independent of $\tau$

[x = 4em, y = 4em,
samples = 500,
domain = 0:2,
ymax = 1.5,
axis x line=bottom,
axis y line=left,
xlabel =  $m$,
ylabel =  $\tau (m)$,
x label style=at=(axis description cs:1,0),anchor=south,
y label style=at=(axis description cs:.07,0.9),rotate=270,anchor=west,
xtick = 1,
xticklabels = $1$,
ytick = 1,
yticklabels = $1$,

[black,domain=0:1.9,line width = 2pt] expression min(x,1);
[black,line width=0pt] expression 0;

$-\partial(-\tau)(1)= [0,1]$

[x = 4em, y = 4em,
samples = 500,
domain = 0:2,
ymax = 1.5,
axis x line=bottom,
axis y line=left,
xlabel = $b$,
ylabel = $\varepsilon (b)$,
x label style=at=(axis description cs:1,0),anchor=south,
y label style=at=(axis description cs:.1,0.9),rotate=270,anchor=west,
xtick = 1,
xticklabels = $1$,
ytick = 1,
yticklabels = $1$,
[black,domain=0:1.9,line width = 2pt] expression max(0,1-x);
[black,line width=0pt] expression 0;

$b+\varepsilon(b)\equiv 1$ for $b\in [0,1]$

Sketch illustrating <ref>. A kink in $\tau$ leads to multiple optimal friction coefficients.

§ ACKNOWLEDGEMENTS

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the priority program SPP 1962, grant WI 4654/1-1, and under Germany's Excellence Strategy EXC 2044 – 390685587, Mathematics Münster: Dynamics–Geometry–Structure. B.W.’s and J.L.'s research was supported by the Alfried Krupp Prize for Young University Teachers awarded by the Alfried Krupp von Bohlen und Halbach-Stiftung. The publication was supported by the Open Access Publication Fund of the University of Münster.

Missing 'biblatex' package
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Birkhäuser Basel

seriesLectures in Mathematics. ETH Zürich
subtitleIn Metric Spaces and in the Space of Probability Measures
titleGradient Flows

titleVisualization of genomic data with the Hilbert curve

journaltitleESAIM: Control, Optimisation and Calculus of Variations
titleOptimal networks for mass transportation problems

journaltitlePublicacions Matemàtiques
titleTraffic Plans

seriesLecture notes in Mathematics
titleOptimal Transportation Networks

New York

John Wiley & Sons, Inc.

seriesWiley Series in Probability and Statistics
titleConvergence of probability measures

journaltitleAnnals of Statistics
titleMeasurable Selections of Extrema

subtitlePhD thesis
titleOptimization problems for transportation networks

Vieweg+Teubner Verlag

titleFelix Hausdorff zum Gedächtnis - Band I

Berlin Heidelberg

titleOptimal Urban Networks via Mass Transportation

journaltitleJ. Math. Pures Appl. (9)
titleEquivalent formulations for the branched transport and urban planning problems

journaltitleCalc. Var. Partial Differential Equations
titleGeneral transport problems with branched minimizers as functionals of 1-currents with prescribed boundary
pagesPaper No. 8239

journaltitleAdv. Calc. Var.
titleA phase-field approximation of the Steiner problem in dimension two

journaltitleReal Analysis Exchange
titleFine variation and fractal measures

Berlin, Heidelberg

Springer Spektrum

titleMaß- und Integrationstheorie

Cambridge University Press

titleThe geometry of fractal sets

journaltitleCalc. Var. Partial Differential Equations
titlePhase field approximations of branched transportation problems
pagesPaper No. 3734

New York

seriesDie Grundlehren der mathematischen Wissenschaften
titleGeometric measure theory

journaltitleThe American Mathematical Monthly
titleA Proof of Lusin's Theorem

journaltitleProceedings of the American Mathematical Society
titleRectifiable Metric Spaces: Local Structure and Regularity of the Hausdorff Measure

Reading MA

seriesAddison-Wesley Series in Mathematics
titleReal Analysis

Cambridge University Press

titleGeometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability

journaltitleBulletin of the American Mathematical Society
titleTransformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems

journaltitleAdvanced Nonlinear Studies
titleSynchronic and Asynchronic Descriptions of Irrigation Problems

journaltitleInterfaces and Free Boundaries
titleA variational model of irrigation patterns

journaltitleCalculus of Variations and Partial Differential Equations
titleExistence and regularity results for the Steiner problem

Springer International Publishing

titleCalculus of variations

New York

Macmillan Publishing Company

titleReal analysis

Birkhäuser Verlag

subtitleCalculus of Variations, PDEs, and Modeling
titleOptimal Transport for Applied Mathematicians

subtitleIn: Mathematical Modelling of Bodies with Complicated Bulk and Boundary Behavior
titleDivergence measure vectorfields: their structure and the divergence theorem

subtitleLecture notes
titleIntroduction to Geometric Measure Theory

journaltitleAlgebra i Analiz
titleDecomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows

Annals of Mathematics

abstractWe prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite-mass flat chain to be rectifiable.
journaltitleAnnals of Mathematics
titleRectifiability of Flat Chains

journaltitleCalc. Var. Partial Differential Equations
titlePhase field models for two-dimensional branched transportation problems
pagesPaper No. 16431

journaltitleCommunications in Contemporary Mathematics
titleOptimal paths related to transport problems

journaltitleCalc. Var. Partial Differential Equations
titleInterior regularity of optimal transport paths