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arXiv 1908.114712019-08math.MGmath.CA

Menger曲率与测度的$C^{1,α}$可求长性

Menger curvatures and $C^{1,α}$ rectifiability of measures

Silvia Ghinassi, Max Goering

备注：: To appear in Archiv der Mathematik
DOI：: 10.1007/s00013-019-01414-6

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AI总结：

本文通过推进$β$数与离散曲率的关系，证明弱密度条件下逐点离散曲率的几乎处处有限性可推出测度的$C^{1,α}$ $n$-可求长性。

摘要：

我们进一步发展$β$数与离散曲率之间的关系，以给出一个新证明：在弱密度假设下，对$\mathbb{R}^{m}$中$μ$-几乎处处的$x$，逐点离散曲率$\operatorname{curv}^α_{μ;2}(x,r)$有限，蕴含$μ$是$C^{1,α}$ $n$-可求长的。

英文摘要：

We further develop the relationship between $β$-numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature $\operatorname{curv}^α_{μ;2}(x,r)$ at $μ$- a.e. $x \in \mathbb{R}^{m}$ implies that $μ$ is $C^{1,α}$ $n$-rectifiable.

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