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, 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: 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.