Menger curvature and \(C^{1}\) regularity of fractals (Q2706591)

The Menger curvature \(c(x,y,z)\) of three points \(x\), \(y\) and \(z\) in the plane is the reciprocal of the radius of the circle passing through these points. Integrals of the Menger curvature can be used to measure smoothness properties of sets. Along this line, the authors show that if \(E\) is an \(s\)-regular subset of the plane for which the triple integral NEWLINE\[NEWLINE\int_E \int_E \int_E c(x,y,z)^{2s} d{\mathcal H}^s xd{\mathcal H}^s yd{\mathcal H}^szNEWLINE\]NEWLINE is finite and if \(0< s\leq 1/2\), then \({\mathcal H}^s\)-almost all of \(E\) can be covered with countably many \(C^1\) curves. As an example shows, this is not the case for \(1/2< s< 1\). For \(s=1\), this is again true, due to \textit{J. C. Léger} [Ann. Math. (2) 149, No. 3, 831-869 (1999; preceding review)].