Midpoint convex functions majorized by midpoint concave functions
The main result is: if \(f(x)\leq g(x),\) \(f[(x+y)/2]\leq [f(x)+f(y)]/2,\) \(g[(x+y)/2]\geq [g(x)+g(y)]/2\) for all x,y on an open convex subset D of \({\mathbb{R}}^ n\), then there exist functions a,h,k additive, convex or concave, respectively, such that \(f(x)=a(x)+h(x),g(x)=a(x)+k(x)\) on D. If \(D={\mathbb{R}}^ n\) then h and k are constant. - There is also a somewhat similar statement on set-valued midpoint-convex functions (without majorant condition), and generalizations to arbitrary vector spaces are mentioned.
scientific article; zbMATH DE number 3989677