  We generalize the notion of integral Menger curvature introduced by Gonzalez
and Maddocks 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-Slobodeckij spaces. Moreover, 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.
Consequently, $intM^{p,q}$ 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
$C^{\infty}$-smoothness of critical points.
