2 problems

Let $(M,V,\varepsilon)$ be a closed connected weighted marked surface, where $\varepsilon:V\rightarrow\{0,1\}$, $\chi(M)=0$, and $d$ is a piecewise linear metric on…

Let $X_p$ be the doubled regular $p$-gon, where $p$ is an odd prime. The minimizing index $\operatorname{minind}(X_p)$ is the smallest positive integer $k$ such that $X_p$ admits a…