Majorization by hemispheres and quadratic isoperimetric constants

Abstract

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.

Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

“…Proof of (27). By (25), [BI02] and [Cre20] one has 1 4π ≤ C A (X) ≤ 1 2π for every nontrivial Banach space X and by Theorem 1.4 the interval [ 1 4π , 1 2π ) is contained in QIS A (Ban). Thus the proof is completed by Remark 4.10 and noting that C A (R) = 0.…”

“…and let µ p be the measure on S 1 which is absolutely continuous with density h p . The most technical part in the proof of Proposition 3.3 amounts to the following lemma, compare Sections 3.2 and 3.3 in [Cre20].…”

“…The filling area Fill A (γ) of a closed curve γ in X is defined as the infimum of A(f ) where f ranges over all Lipschitz discs spanning γ. The main result of [Cre20] states: if γ ∶ (S 1 , d S 1 ) → X is 1-Lipschitz then γ extends to a 1-Lipschitz map m ∶ H 2 → X.…”

“…A somewhat analytical argument involving the optimal transport going on in the proof of the main result of [Cre20] then shows that…”

Space of minimal discs and its compactification

“…It turns out that a geodesic metric disc Z satisfies a C-quadratic isoperimetric inequality iff every Jordan domain U ⊂ Z satisfies H 2 (U ) ≤ C • l(∂U ) 2 , see [5,17]. For example compact Riemannian and more generally Finsler manifolds, CAT(κ)-spaces and Banach spaces satisfy some quadratic isoperimetric inequality, see [4,13].…”