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.
"Isoperimetric characterization of upper curvature bounds." Acta Math. 221 (1) 159 - 202, September 2018. https://doi.org/10.4310/ACTA.2018.v221.n1.a5