Affine subspaces of curvature functions from closed planar curves

Given a pair of real functions $(k,f)$, we study the conditions they must satisfy for $k+\lambda f$ to be the curvature in the arc-length of a closed planar curve for all real $\lambda$. Several equivalent conditions are pointed out, certain periodic behaviours are shown as essential and a family of such pairs is explicitely constructed. The discrete counterpart of the problem is also studied. Finally, the characterization obtained is used to show that a sufficient analogue of the 4-vertex theorem cannot be developed.