I see Beurling’s extremality criterion at two places: the proof is almost identical, but the statement is very different. Below,
$$
\ell_\rho (\gamma) = \int_\gamma \rho(z) |dz|.
$$
"Extremal" means to maximise the value
$$
\sup_{\rho} \frac{\inf_{\gamma \in\Gamma} \ell_\rho(\gamma)^2}{\int_X \rho^2|dz|^2} 
$$
Here, $\Gamma$ is a family of paths on a Riemann surface $X.$


Firstly, in [*Riemann surfaces, dynamics and
geometry* by McMullen](http://people.math.harvard.edu/~ctm/home/text/class/harvard/275/09/html/base/rs/rs.pdf), on page 22, there is the following theorem:

> Suppose a measure $\rho^2$ on $X$ lies in the closed convex hull of the measures $$ \{\rho|\gamma : \gamma \in \Gamma, \ell_\rho (\gamma) = \ell_\rho (\Gamma)\}.$$ Then $\rho$ is extremal for $\Gamma.$



Then we have another different statement of the theorem in [these graph theory notes](https://www.math.ksu.edu/~pietro/Research/beurling-kkt.pdf): I have restated the theorem so that it becomes a result for Riemann surfaces (instead of graph theory)

> A density $\rho$ is extremal if there is $\Gamma_0 \subseteq \Gamma$ satisfying $\ell_\rho(\gamma)=1$ for $\gamma \in \Gamma_0$, such that $\int_X h\rho \geq 0$ whenever $h:X\to \mathbb R$ is a function satisfying $\ell_h(\gamma) \geq 0,\forall\gamma \in \Gamma_0.$

The first statement of the theorem does not make sense for me. **What does $\rho|\gamma$ mean? $\rho$ restricted to $\gamma$? If this is just a restriction, then what does "convex hull" mean?** **Why is the condition named "convex"?** After all the statement of the second theorem does not look the same as the usual definition of convexity.

An explanation on these unfamiliar notation/terminology would be really helpful.