Deforming Locally Convex Curves into Curves of Constant k-order Width

A nonlocal curvature flow is introduced to evolve locally convex curves in the plane. It is proved that this flow with any initial locally convex curve has a global solution, keeping the local convexity and the elastic energy of the evolving curve, and that, as the time goes to infinity, the curve converges to a smooth, locally convex curve of constant k-order width. In particular, the limiting curve is a multiple circle if and only if the initial locally convex curve is k-symmetric.