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.

9 pages