Absolutely minimising Lipschitz extensions between Hilbert spaces
Abstract
Every Lipschitz map from a closed subset of a real Hilbert space into a real Hilbert space has an absolutely minimising Lipschitz extension: an extension whose Lipschitz constant on every open subset of the complement, bounded or not, equals the Lipschitz constant of its restriction to the boundary of that subset. Neither space need be separable. This answers a question of Naor and Sheffield. For targets of dimension at least two we prove a density theorem on connected domains: every optimal extension of a boundary datum can be approximated, uniformly on compact sets, by extensions whose Lipschitz constant on every nonempty open subset, and on its boundary, equals the constant of the datum. Consequently, in contrast with the scalar case, such a datum has a unique absolutely minimising extension exactly when its optimal extension is unique, and absolutely minimising extensions need not be closed under uniform convergence, even for fixed smooth data on a planar disk. The extensions are built by finite piecewise isometric interpolation: on nested polyhedra with dense union they are piecewise affine and preserve distances in at least two directions on each simplex, a structure described through leaves of 1-Lipschitz maps. The boundary constraints are handled by a midpoint argument for admissible values, and a Kuratowski-Zorn argument removes the separability assumption while preserving previously prescribed values. We also characterise the bounded linear maps that are absolutely minimising extensions of all data determining their operator norm.
Related
- Examples of absolutely minimal polyhedral chains in maximum norm spaces
- Hilbert's basis theorem for Poisson Ore extensions
- Maximal Spaceability of the set of non-absolutely summing operators in quasi-Banach spaces
- Absolutely summing Toeplitz operators on weighted Fock spaces
- Absolutely summing operators on Bergman spaces over convex domains of finite type
BibTeX
@misc{ciosmak2026absolutely,
title = {{Absolutely minimising Lipschitz extensions between Hilbert spaces}},
author = {Krzysztof J. Ciosmak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17194},
eprint = {2610.17194},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17194},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Absolutely minimising Lipschitz extensions between Hilbert spaces
AU - Krzysztof J. Ciosmak
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.17194
UR - https://arxiv.org/abs/2610.17194
AB - Every Lipschitz map from a closed subset of a real Hilbert space into a real Hilbert space has an absolutely minimising Lipschitz extension: an extension whose Lipschitz constant on every open subset of the complement, bounded or not, equals the Lipschitz constant of its restriction to the boundary of that subset. Neither space need be separable. This answers a question of Naor and Sheffield. For targets of dimension at least two we prove a density theorem on connected domains: every optimal extension of a boundary datum can be approximated, uniformly on compact sets, by extensions whose Lipschitz constant on every nonempty open subset, and on its boundary, equals the constant of the datum. Consequently, in contrast with the scalar case, such a datum has a unique absolutely minimising extension exactly when its optimal extension is unique, and absolutely minimising extensions need not be closed under uniform convergence, even for fixed smooth data on a planar disk. The extensions are built by finite piecewise isometric interpolation: on nested polyhedra with dense union they are piecewise affine and preserve distances in at least two directions on each simplex, a structure described through leaves of 1-Lipschitz maps. The boundary constraints are handled by a midpoint argument for admissible values, and a Kuratowski-Zorn argument removes the separability assumption while preserving previously prescribed values. We also characterise the bounded linear maps that are absolutely minimising extensions of all data determining their operator norm.
ER -