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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.

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