  Relatively extremal knots are the relative minima of the ropelength
functional in C^1 topology. On the set curves of fixed length, they are the
relative maxima of thickness (normal injectivity radius) functional, including
the ideal knots. We prove that a C^{1,1} relatively extremal knot in R^n has
thickness equal to half of the minimal double critical distance unless it has
constant maximal (generalized) curvature. This result also applies to links
since our method is local. Our main approach is to show that the shortest
curves with bounded curvature and C^1 boundary conditions in R^n contain CLC
(circle-line-circle) curves, if they do not have constant maximal curvature.
