Let $S_{g}$ be the closed orientable genus $g$ surface. I'm interested in studying a certain class $\Gamma$ of graphs on $S_{g}$ which fill (the complementary regions are simply connected), and which hyperbolic metrics on $S_{g}$ optimize the total length of a graph in $\Gamma$ (I'd be happy to make this more explicit).
While thinking about this, the following question arose:
Let $\mu=(\mu_{1},...,\mu_{n})\in \mathbb{N}^{n}$, and define $\rho_{(\mu)}: \mathbb{(R_{>0})}^{n}\rightarrow \mathbb{R}$ to be the "perimeter function" for $\mu$, which takes as input an $n$-tuple of areas $(\alpha_{1},...,\alpha_{n})$ subject to the constraint
$\sum_{i=1}^{n} \alpha_{i}= 2\pi(2g-2)$, (the area of any hyperbolic surface with topological type $S_{g}$)
and outputs the sum of the perimeters of a collection of $n$ regular polygons, such that the $j^{th}$ polygon has $\mu_{j}$ sides and area $\alpha_{j}$.
$\textbf{Question: For which choice(s) of $(\alpha_{1},...,\alpha_{n})$ is $\rho_{(\mu)}$ minimized? How does this depend on $\mu$?} $
My first guess was that $\rho_{(\mu)}$ would be minimized by setting all but one area equal to $0$; in other words, put all the area into one polygon, with $\max_{j}(\mu_{j})$ number of sides. But after a bit of experimenting, I think I've convinced myself that there exists choices of $\mu$ for which this fails.
I'm mostly interested in the special case where there is a relationship between $n$ (the number of polygons) and $\sum_{j} \mu_{j}$ (the total number of sides). Specifically, $n= \frac{\sum_{j}\mu_{j}}{4}-2g+2$. In this setting, I'd like to show that the naive guess is right; that is, that the perimeter is optimized by a single regular, right-angled $(8g-4)$-gon.
I've tried looking around to see if people have addressed questions like these and I couldn't find much of anything, although I'm not very familiar with this body of literature.
Any help would be greatly appreciated, and thank you for reading.
$\textbf{UPDATE}:$ I thought it might be useful to give an example of a $\mu$ for which the naive guess is false; suppose $\mu=(k,k,k,...,k)$ for some large $k$, and let $f_{k}(x)$ be the perimeter of a regular hyperbolic $k$-gon with area $x$. The fundamental issue is that $f_{k}(x)$ is not concave-down, unlike in Euclidean geometry where the perimeter of a regular $k$-gon is expressible as some square root of the area. Indeed, one can write down an explicit formula for $f_{k}(x)$ using the usual hyperbolic trig formulas, and the second derivative is positive for $x$ sufficiently large (and of course, smaller than the maximum possible area of a regular $k$-gon). In particular, this means that for certain values of $x$, the most efficient way to enclose an area of $x$ using regular $k$-gons is $\textit{not}$ to use a single $k$-gon. At first, I found this somewhat surprising, but then I realized this is obvious if you choose $x$ to be the area of an ideal $k$-gon.
