h-PRINCIPLES FOR CURVES AND KNOTS OF CONSTANT CURVATURE

We prove that C∞ curves of constant curvature satisfy, in the sense of Gromov, the relative C-dense h-principle in the space of immersed curves in Euclidean space Rn≥3. In particular, in the isotopy class of any given C knot f there exists a C∞ knot e f of constant curvature which is C-close to f . More importantly, we show that if f is C, then the curvature of e f may be set equal to any constant c which is not smaller than the maximum curvature of f . We may also require that e f 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.