The paper studies \(\frac{1}{k}\)-geodsics in doubled regular \(n\)-gons \(X_n\). The class of \(\frac{1}{k}\)-geodesics was introduced in [C. Sormani, Adv. Math. 213, No. 1, 405–439 (2007; Zbl 1124.53019)]. Therein the minimizing index of a \(\frac{1}{k}\)-geodesic was defined as the minimal \(k\in \mathbb{N}\) such that a given closed geodesic is a \(\frac{1}{k}\)-geodesic.
The results address the minimal index of the \(X_n\)’s and \(X_\infty\), the Gromov-Hausdorff limit of \((X_n)_{n\in\mathbb{N}}\), respectively. The minimal index of \(X_n\) is the minimum over all indices of \(\frac{1}{k}\)-geodesics. Connections to convex billiards and the systolic problem are discussed.