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 $\mathrm{Lip}_0(M)$ 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: $\mathrm{Lip}_0(M)$ has a strongly unique predual whenever $M$ is a convex subset of a finite-dimensional normed space.