h-Principles for curves and knots of constant curvature

We prove that C-infinity curves of constant curvature satisfy, in the sense of Gromov, the relative C-1-dense h-principle in the space of immersed curves in Euclidean space R-n>=3. In particular, in the isotopy class of any given C-1 knot f there exists a C-infinity knot (f) over tilde of constant curvature which is C-1-close to f. More importantly, we show that if f is C-2, then the curvature of (f) over tilde may be set equal to any constant c which is not smaller than the maximum curvature of f. We may also require that (f) over tilde be tangent to f along any finite set of prescribed points, and coincide with f over any compact set with an open neighborhood where f has constant curvature c. The proof involves some basic convexity theory, and a sharp estimate for the position of the average value of a parameterized curve within its convex hull.