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Erratum and a counterexample to: Stability of diametral diameter two properties
Abstract
In Proposition 3.5 of our paper [J. Langemets and K. Pirk, Stability of diametral diameter two properties, RACSAM (2021)], we claimed that the convex DLD2P lifts from an M-ideal to its superspace. The proof contains a quantifier error. We show that its valid part yields the following replacement: for every ε>0, every slice of the unit ball of the superspace contains a (2−ε) Δ-point of norm arbitrarily close to one. In particular, the unit ball is the closed convex hull of these points at each fixed scale, and the superspace has the local diameter two property. Finally, we prove that the original claim was false, we construct Banach spaces Y and X such that Y has the convex DLD2P, Y is an M-ideal in X, yet X fails the convex DLD2P.
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BibTeX
@misc{langemets2026erratum,
title = {{Erratum and a counterexample to: Stability of diametral diameter two properties}},
author = {Johann Langemets},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28569},
eprint = {2609.28569},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28569},
}
RIS
TY - UNPB
TI - Erratum and a counterexample to: Stability of diametral diameter two properties
AU - Johann Langemets
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28569
UR - https://arxiv.org/abs/2609.28569
AB - In Proposition 3.5 of our paper [J. Langemets and K. Pirk, Stability of diametral diameter two properties, RACSAM (2021)], we claimed that the convex DLD2P lifts from an M-ideal to its superspace. The proof contains a quantifier error. We show that its valid part yields the following replacement: for every ε>0, every slice of the unit ball of the superspace contains a (2−ε) Δ-point of norm arbitrarily close to one. In particular, the unit ball is the closed convex hull of these points at each fixed scale, and the superspace has the local diameter two property. Finally, we prove that the original claim was false, we construct Banach spaces Y and X such that Y has the convex DLD2P, Y is an M-ideal in X, yet X fails the convex DLD2P.
ER -