Unique predual problem for Lipschitz spaces

The problem asks whether Lipschitz function spaces have unique, or strongly unique, preduals in broad classes of metric spaces. A 2016 paper claimed the result for all $\operatorname{Lip}(X)$ and for $\operatorname{Lip}_0(X)$ when $X$ is finite-dimensional in the relevant convex setting, but part of that proof was later withdrawn.

Known results

• $\operatorname{Lip}(X)$ has a unique predual for every metric space $X$ (Weaver, 2016).
• The earlier finite-diameter and complete-convex claims for $\operatorname{Lip}_0(X)$ relied on a faulty proof of Lemma $3.1$.
• For complete convex $X$, strong uniqueness reduces conditionally to strong uniqueness on every closed ball $X_n$.

September 1, 2026 counterexample and special-case proof

On September 1, 2026, a preprint reported an explicit counterexample to inheritance of strong unique preduals by $1$-codimensional weak-star-closed subspaces and claimed a new proof for the convex finite-dimensional case. This corrects a proof technique but does not settle the problem for all complete metric spaces.

Current status (as of September 2026): uniqueness for $\operatorname{Lip}(X)$ is retained, while the broad strongly unique-predual claim for $\operatorname{Lip}_0(X)$ remains open; the finite-dimensional convex result is claimed but unverified.