A note on integral Menger curvature for curves

For continuously differentiable embedded curves gamma, we will give a necessary and sufficient condition for the boundedness of integral versions of the Menger curvature. We will show that for p > 3 the integral Menger curvature M-p(gamma) is finite if and only if gamma belongs to the Sobolev Slobodeckij space W-2-2/p,W-p (R/Z,R-n). The quantity J(p)(gamma) - defined by taking the supremum of the p-th power of Menger curvature with respect to one variable and then integrating over the remaining two - is finite for p > 2, if and only if gamma belongs to the space W-2-1/p,W-p. (C) 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim