The unique predual problem for Lipschitz spaces, revisited
Abstract
In his 2018 paper ``On the unique predual problem for Lipschitz spaces'', N. Weaver published proofs that Banach spaces Lip0(M) of Lipschitz functions on a complete metric space M have strongly unique preduals whenever M has finite diameter or is geodesic. A gap was recently noticed in the proof of a crucial lemma that claimed that the property of having a strongly unique predual passes to 1-codimensional weak*-closed subspaces. In this note, we confirm that the lemma is actually false by providing an explicit counterexample. We also expand on some of Weaver's original arguments to provide a new, valid proof of the following particular case: Lip0(M) has a strongly unique predual whenever M is a convex subset of a finite-dimensional normed space.
Related
- Lipschitz-free spaces are strongly unique preduals
- Improved Bounds For The Lipschitz Retraction Constant In Classical Banach Spaces
- Non-compact convex hulls in CAT(0) spaces revisited
- Isometric copies of ℓ∞n in Lipschitz-free spaces over finite metric spaces
- Lipschitz clustering in normed spaces and barycentric spaces
BibTeX
@misc{aliaga2026unique,
title = {{The unique predual problem for Lipschitz spaces, revisited}},
author = {Ramón J. Aliaga and Felipe Vico},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00970},
eprint = {2609.00970},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00970},
}
RIS
TY - UNPB
TI - The unique predual problem for Lipschitz spaces, revisited
AU - Ramón J. Aliaga
AU - Felipe Vico
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00970
UR - https://arxiv.org/abs/2609.00970
AB - In his 2018 paper ``On the unique predual problem for Lipschitz spaces'', N. Weaver published proofs that Banach spaces Lip_0(M) of Lipschitz functions on a complete metric space M have strongly unique preduals whenever M has finite diameter or is geodesic. A gap was recently noticed in the proof of a crucial lemma that claimed that the property of having a strongly unique predual passes to 1-codimensional weak^*-closed subspaces. In this note, we confirm that the lemma is actually false by providing an explicit counterexample. We also expand on some of Weaver's original arguments to provide a new, valid proof of the following particular case: Lip_0(M) has a strongly unique predual whenever M is a convex subset of a finite-dimensional normed space.
ER -