The Dehn function, or more precisely its asymptotic growth, is a well-studied quasi-isometry invariant of a finitely generated group. In the talk we will discuss a metric version of the Dehn function which allows to characterize notions of non-positive curvature such as Gromov hyperbolicity or the CAT(0) condition. The main result will be a sharp upper bound on the Dehn functions of Banach spaces. Its proof relies on a majorization result in the spirit of Reshetnyak's theorem for CAT(0) spaces.