Geometric characterization of dense strongly norm-attaining Lipschitz functions

Characterize geometrically all metric spaces $M$ for which the set of strongly norm-attaining scalar-valued Lipschitz functions $\SNA(M)$ is dense in the Banach space of scalar-valued Lipschitz functions $(M)$.

Background

The paper contrasts its complete scalar-valued characterization for strong minimum-attainment with the corresponding strong norm-attainment problem. Several classes of metric spaces are known for which density fails, and positive results follow from properties of the Lipschitz-free space, but no complete geometric description is currently available.

References

Nevertheless, no complete geometric characterization of those metric spaces $M$ for which $\SNA(M,Y)$ is dense in $(M,Y)$ is currently known, even when $Y=R$.

— A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps  (2609.11080 - Choi, 10 Sep 2026) in Section 1, Introduction