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Mathematics, Classical Analysis and ODEs, math.CA,
Abstract:
For an $m$~dimensional $\mathcal{H}^m$~measurable set $\Sigma$ we define,
axiomatically, a class of Menger like curvatures $\kappa : \Sigma^{m+2} \to
[0,\infty)$ which imitate, in the limiting sense, the classical curvature if
$\Sigma$ is of class~$\mathscr{C}^2$. With each $\kappa$ we associate an
averaged curvature $\mathcal{K}^{l,p}_{\kappa}[\Sigma] : \Sigma \to [0,\infty]$
by integrating $\kappa^p$ with respect to $l-1$ parameters and taking supremum
with respect to $m+2-l$ parameters. We prove that if $\Sigma$ is a~priori
$(\mathcal{H}^m,m)$~rectifiable (of class $\mathscr{C}^1$) with
$\mathcal{H}^m(\Sigma) < \infty$ and $\mathcal{K}^{l,p}_{\kappa}[\Sigma](a) <
\infty$ for $\mathcal{H}^m$ almost all $a \in \Sigma$, then $\Sigma$ is in fact
$(\mathcal{H}^m,m)$~rectifiable of class~$\mathscr{C}^{1,\alpha}$, where
$\alpha = 1 - m(l-1)/p$. We also prove an analogous result for the
tangent-point curvature and we show that $\alpha$ is sharp.
