Acta Mathematica

Volume 221 (2018)

Number 1

Isoperimetric characterization of upper curvature bounds

Pages: 159 – 202



Alexander Lytchak (Mathematisches Institut, Universit¨at K¨oln, Germany)

Stefan Wenger (Department of Mathematics, University of Fribourg, Switzerland)


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.

Received 9 December 2017

Accepted 3 August 2018

Published 6 November 2018