A Sharp Lower Bound for Integral Menger Curvature

We prove that round circles $C_L$ are the unique minimisers of integral Menger curvature among embedded closed $C^1$ curves $γ$ of prescribed length $L$ for every $p\geq 2$, i.e. $M_p(C_L) \leq M_p(γ)$, with equality if and only if $γ$ parametrises a round circle. The proof combines a three-point sy