A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps
Abstract
We completely characterize the denseness of strongly minimum-attaining Lipschitz functions, a minimum analogue for strongly norm-attaining Lipschitz functions, in terms of bi-Lipschitz embeddings. More precisely, our main result shows that the set of strongly minimum-attaining Lipschitz functions defined on a complete metric space M fails the denseness if and only if M is bi-Lipschitz equivalent to a subset of ℝ with positive Lebesgue measure, or equivalently, if M admits a bi-Lipschitz embedding into ℝ and M has positive 1-dimensional Hausdorff measure. As a consequence, we provide an isometric characterization of the pure 1-unrectifiability of M in terms of strongly minimum-attaining Lipschitz maps defined on bi-Lipschitz copies of closed subsets of M. Several counterexamples showing that the main result cannot be naturally extended to the vector-valued setting are also presented.
Related
- Norm and Minimum Attainment of Foguel-Type Operators
- Low regularity maps on Lipschitz manifolds: lifting, pullback of forms, and approximation
- Möbius Maps, Reflections and Lipschitz Constants
- Invariant subspaces for free linearizations of Lipschitz maps
- Lipschitz regularity of Sobolev maps from RCD(K,N) spaces to CAT(0) spaces
BibTeX
@misc{choi2026bilipschitz,
title = {{A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps}},
author = {Geunsu Choi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11080},
eprint = {2609.11080},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11080},
}
RIS
TY - UNPB
TI - A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps
AU - Geunsu Choi
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11080
UR - https://arxiv.org/abs/2609.11080
AB - We completely characterize the denseness of strongly minimum-attaining Lipschitz functions, a minimum analogue for strongly norm-attaining Lipschitz functions, in terms of bi-Lipschitz embeddings. More precisely, our main result shows that the set of strongly minimum-attaining Lipschitz functions defined on a complete metric space M fails the denseness if and only if M is bi-Lipschitz equivalent to a subset of ℝ with positive Lebesgue measure, or equivalently, if M admits a bi-Lipschitz embedding into ℝ and M has positive 1-dimensional Hausdorff measure. As a consequence, we provide an isometric characterization of the pure 1-unrectifiability of M in terms of strongly minimum-attaining Lipschitz maps defined on bi-Lipschitz copies of closed subsets of M. Several counterexamples showing that the main result cannot be naturally extended to the vector-valued setting are also presented.
ER -