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An unbounded convex set with the fixed point property for nonexpansive mappings
Abstract
We construct an equivalent norm on ℝ⊕ℓ2 and a nonempty closed convex unbounded subset having the fixed point property for nonexpansive self-mappings. The ambient renormed space has the fixed point property on bounded closed convex subsets. The main point is a geometric localization estimate at a distinguished point of the unbounded set; the fixed point conclusion then follows from the Kirk--Ray localization theorem.
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BibTeX
@misc{japon2026unbounded,
title = {{An unbounded convex set with the fixed point property for nonexpansive mappings}},
author = {M. Japón and Javier Laguna-Ricart},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14868},
eprint = {2610.14868},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14868},
}
RIS
TY - UNPB
TI - An unbounded convex set with the fixed point property for nonexpansive mappings
AU - M. Japón
AU - Javier Laguna-Ricart
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2610.14868
UR - https://arxiv.org/abs/2610.14868
AB - We construct an equivalent norm on ℝ⊕ℓ_2 and a nonempty closed convex unbounded subset having the fixed point property for nonexpansive self-mappings. The ambient renormed space has the fixed point property on bounded closed convex subsets. The main point is a geometric localization estimate at a distinguished point of the unbounded set; the fixed point conclusion then follows from the Kirk--Ray localization theorem.
ER -