Undistorted fillings in subsets of metric spaces

Abstract

We prove that if a quasiconvex subset X of a metric space Y has finite Nagata dimension and is Lipschitz k-connected or admits Euclidean isoperimetric inequalities up to dimension k for some k then X is isoperimetrically undistorted in Y up to dimension k + 1. This generalizes and strengthens a recent result of the third named author and has several consequences and applications. It yields for example that in spaces of finite Nagata dimension, Lipschitz connectedness implies Euclidean isoperimetric inequalities, and Euclidean isoperimetric inequalities imply coning inequalities. It furthermore allows us to prove an analog of the Federer-Fleming deformation theorem in spaces of finite Nagata dimension admitting Euclidean isoperimetric inequalities.

Geometric and analytic structures on metric spaces homeomorphic to a manifold

“…Finally, let 0, 1 × S denote the product current (see [4,Section 3.3] for the definition), and note that this is an element of I n+1 ([0, 1] × ℓ ∞ ). Let furthermore H : [0, 1] × C → ℓ ∞ be the straight-line homotopy from the identity on C to ψ.…”

“…[52, Theorem 2.9], that R = H # ( 0, 1 ×S ) ∈ I n+1 (ℓ ∞ ) satisfies ∂R = S . Moreover, [4,Lemma 3.5] shows that…”

“…For every simplex σ ⊂ Σ and every vertex v ∈ σ we have d(y, h(v)) ≤ Cd(y, X) for all y ∈ g −1 (st σ), where C > 0 only depends on the data of X. The relevant notation concerning simplicial complexes can be found in [4,Section 2.2]. We denote by Λ ≥ 1 the linear local contractibility constant of X and set λ ≔ [3C(2Λ) k+2 ] −1 .…”

“…One can show that when n = 2 our Theorem 1.1 remains true if the linear local contractibility is replaced by the condition that X be quasiconvex. The proof does not rely on Bate's theorem or (1.1), and requires only techniques from [4] and [39]. When n ≥ 3 we do not know whether the theorem remains true without the linear local contractibility condition.…”