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+λf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+\lambda f$$\end{document} to be the curvature in the arc-length of a closed planar curve for all real λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document}. 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.