For which positive $p$ is the integral Menger curvature $\mathcal{M}_{p}$ finite for all simple polygons?

Abstract

In this brief note we show that the integral Menger curvature $\mathcal{M}_{p}$ is finite for all simple polygons if and only if $p\in (0,3)$. For the intermediate energies $\mathcal{I}_{p}$ and $\mathcal{U}_{p}$ we obtain the analogous result for $p\in (0,2)$ and $p\in (0,1)$, respectively.