Density characterization for vector-valued strongly minimum-attaining Lipschitz maps

Determine for which complete metric spaces $M$ and Banach spaces $Y$ the set of strongly minimum-attaining Lipschitz maps $\SMA(M,Y)$ is dense in the Banach space $(M,Y)$ of $Y$-valued Lipschitz maps.

Background

The paper proves a complete density characterization for scalar-valued strongly minimum-attaining Lipschitz functions, but shows that the analogous proposed equivalence fails for vector-valued maps. In particular, the density behavior can depend substantially on the geometry of the range space: it holds for certain Hilbert-valued maps and fails for finite-dimensional ℓ∞n\ell_\infty^n-valued maps. The authors therefore leave the general range-dependent characterization unresolved.

References

Motivated by the preceding discussion, we conclude the paper with the following general question.

For which metric spaces $M$ and Banach spaces $Y$ is $\SMA(M,Y)$ dense in $(M,Y)$?

— A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps  (2609.11080 - Choi, 10 Sep 2026) in Section 3, concluding paragraph