Functional inequalities in the framework of Banach spaces

A quadrilateral inequality established by C. Sch\"otz in the context of Hilbert spaces is extended to the framework of Banach spaces. Our approach is based on the theory of majorization and a substitute for the parallelogram law associated with Clarkson's notion of von Neumann-Jordan constant. Several examples (concerning matrix theory and geometry of Banach spaces) that illustrate our new results are also included.