Isoperimetric characterization of upper curvature bounds
  We prove that a proper geodesic metric space has non-positive curvature in the sense of Alexandrov if and only if it satisfies the Euclidean isoperimetric inequality for curves. Our result extends to non-geodesic spaces and non-zero curvature bounds. 
Alexander Lytchak, Stefan Wenger
math.DG math.MG