A relation of thermodynamic relevance between the superadditivity, concavity and homogeneity properties of real-valued functions

We provide a necessary and sufficient condition for the validity of the following Landsberg-Thirring theorem: for a real-valued function on a convex set, any two of the properties of superadditivity, concavity and homogeneity implies the third. Applications to statistical thermodynamics, following Thirring and Landsberg, are briefly reviewed.