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

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

2012-02-02

arxiv.org/abs/1202.0504v1

Mathematics

Classical Analysis and ODEs

9 pages

Scientific paper

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.

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