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 γ ∶ 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 2π . 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.

We investigate the class of geodesic metric discs satisfying a uniform quadratic isoperimetric inequality and uniform bounds on the length of the boundary circle. We show that the closure of this class as a subset of Gromov-Hausdorff space is intimately related to the class of geodesic metric disc retracts satisfying comparable bounds. This kind of discs naturally come up in the context of the solution of Plateau’s problem in metric spaces by Lytchak and Wenger as generalizations of minimal surfaces.

We prove that any length metric space homeomorphic to a surface may be decomposed into non-overlapping convex triangles of arbitrarily small diameter. This generalizes a previous result of Alexandrov-Zalgaller for surfaces of bounded curvature.

We give a solution of Plateau's problem for singular curves possibly having self-intersections. The proof is based on the solution of Plateau's problem for Jordan curves in very general metric spaces by Alexander Lytchak and Stefan Wenger and hence works also in a quite general setting. However the main result of this paper seems to be new even in R n .

Our main result gives an improved bound on the filling areas of closed curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu's classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds.