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Journal Article

#### Minimal Hölder regularity implying finiteness of integral Menger curvature

##### Fulltext (public)

1111.1141.pdf

(Preprint), 561KB

s00229-012-0565-y.pdf

(Any fulltext), 266KB

##### Supplementary Material (public)

There is no public supplementary material available

##### Citation

Kolasinski, S., & Szumańska, M. (2013). Minimal Hölder regularity implying finiteness
of integral Menger curvature.* Manuscripta Mathematica,* *141*(1-2),
125-147. doi:10.1007/s00229-012-0565-y.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-B36A-4

##### Abstract

We study two families of integral functionals indexed by a real number $p >
0$. One family is defined for 1-dimensional curves in $\R^3$ and the other one
is defined for $m$-dimensional manifolds in $\R^n$. These functionals are
described as integrals of appropriate integrands (strongly related to the
Menger curvature) raised to power $p$. Given $p > m(m+1)$ we prove that
$C^{1,\alpha}$ regularity of the set (a curve or a manifold), with $\alpha >
\alpha_0 = 1 - \frac{m(m+1)}p$ implies finiteness of both curvature functionals
($m=1$ in the case of curves). We also show that $\alpha_0$ is optimal by
constructing examples of $C^{1,\alpha_0}$ functions with graphs of infinite
integral curvature.