MAJORIZATION BY HEMISPHERES AND QUADRATIC ISOPERIMETRIC CONSTANTS

Let X be a Banach space or more generally a complete metric space admitting a conical geodesic bicombing. We prove that every closed L-Lipschitz curve gamma : S-1 -> X may be extended to an L-Lipschitz map defined on the hemisphere f : H-2 -> X. This implies that X satisfies a quadratic isoperimetric inequality (for curves) with constant 1/2 pi. We discuss how this fact controls the regularity of minimal discs in Finsler manifolds when applied to the work of Alexander Lytchak and Stefan Wenger.