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We study a two-point self-avoidance energy E q which is defined for all rectifiable curves in R n as the double integral along the curve of 1/r q . Here r stands for the radius of the (smallest) circle that is tangent to the curve at one point and passes through another point on the curve, with obvious natural modifications of this definition in the exceptional, non-generic cases. It turns out that finiteness of E q (γ) for q ≥ 2 guarantees that γ has no self-intersections or triple junctions and therefore must be homeomorphic to the unit circle S 1 or to a closed interval I. For q > 2 the energy E q evaluated on curves in R 3 turns out to be a knot energy separating different knot types by infinite energy barriers and bounding the number of knot types below a given energy value. We also establish an explicit upper bound on the Hausdorff-distance of two curves in R 3 with finite E q -energy that guarantees that these curves are ambient isotopic. This bound depends only on q and the energy values of the curves. Moreover, for all q that are larger than the critical exponent q crit = 2, the arclength parametrization of γ is of class C 1,1−2/q , with Hölder norm of the unit tangent depending only on q, the length of γ, and the local energy. The exponent 1 − 2/q is optimal.Mathematics Subject Classification (2000): 28A75, 49Q10, 53A04, 57M25

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We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a $C^{1,\lambda}$-a-priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale $R>0$ which depends only on an upper bound $E$ for the integral Menger curvature $M_p(\Sigma)$ and the integrability exponent $p$, and \emph{not} on the surface $\Sigma$ itself; below that scale, each surface with energy smaller than $E$ looks like a nearly flat disc with the amount of bending controlled by the (local) $M_p$-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the H\"{o}lder exponent $\lambda$ up to the optimal one, $\lambda=1-(8/p)$, thus establishing a new geometric `Morrey-Sobolev' imbedding theorem. As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature.Comment: 64 pages, 7 figures. Submitted. Version 2: extended comments on the relation to Lerman's and Whitehouse's work on Menger curvature