Towards a regularity theory for integral menger curvature

We generalize the notion of integral Menger curvature introduced by Gonzalez and Maddocks [14] by decoupling the powers in the integrand. This leads to a new two-parameter family of knot energies intM(p,q). We classify finite-energy curves in terms of Sobolev-Slobodeckiı ̆ spaces. More-over, restricting to the range of parameters leading to a sub-critical Euler-Lagrange equation, we prove existence of minimizers within any knot class via a uniform bi-Lipschitz bound. Consequemtly, intM(p,q) is a knot energy in the sense of O’Hara. Restricting to the non-degenerate sub-critical case, a suitable decomposition of the first variation allows to establish a bootstrapping argument that leads to C∞