Minimal Hölder regularity implying finiteness of integral Menger curvature

Abstract

We study two families of integral functionals indexed by a real number $p>0$. One family is defined for 1-dimensional curves in ${\mathbb{R}}^3$ and the other one is defined for $m$-dimensional manifolds in ${\mathbb{R}}^n$. 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.

References in corpus (5)

• Tangent-point self-avoidance energies for curves
• Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds
• High-Dimensional Menger-Type Curvatures-Part II: d-Separation and a Menagerie of Curvatures
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Cited by in corpus (3)

• Geometric Sobolev-like embedding using high-dimensional Menger-like curvature
• Sharp Boundedness and Regularizing effects of the integral Menger curvature for submanifolds
• M{ö}bius-invariant self-avoidance energies for non-smooth sets in arbitrary dimensions