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 R3 and the other one is defined for m-dimensional manifolds in Rn. 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 C1,α regularity of the set (a curve or a manifold), with α> α0 = 1 − m(m+1)p implies finiteness of both curvature functionals (m = 1 in the case of curves). We also show that α0 is optimal by constructing examples of C1,α0 functions with graphs of infinite integral curvature. 1