Quantified compactness in Lipschitz-free spaces of $[-1,1]^n$

Thierry De Pauw

Quantified compactness in Lipschitz-free spaces of $[-1,1]^n$

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arxiv 2504.19100 v1 pith:2FV3ACTG submitted 2025-04-27 math.FA

Thierry De Pauw This is my paper · ORCID

classification math.FA

keywords lipschitz-freecompactnesscurrentsflatspaceallowsboundarycharacterize

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abstract

We show that the members of the Lipschitz-free space of $[-1,1]^n$ are exactly the 0-dimensional flat currents whose "boundary" vanishes. The connection with normal and flat currents allows to use the Federer-Fleming compactness and deformation theorems in this context. We characterize the compact subsets of this Lipschitz-free space and we quantify their $\epsilon$-entropy.

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Structure of Metric $1$-currents: approximation by normal currents and representation results
math.MG 2025-08 conditional novelty 8.0 of 10

Every metric 1-current in a complete quasiconvex metric space is a mass limit of normal 1-currents, and in complete separable spaces it is an integral of curve fragments.

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