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PreprintCC BY 4.0arXiv

The Kadets--Werner modification of Bourgain--Rosenthal space is asymptotically midpoint uniformly convex

  • Florent P. Baudier
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Abstract

Let XKW be the Kadets--Werner modification of a closed subspace of L1 constructed by Bourgain and Rosenthal in 1980. In this short note, it is shown that XKW is asymptotically midpoint uniformly convex. Since XKW has the Daugavet Property, it fails the Point of Continuity Property and thus does not admit an equivalent norm that is asymptotically uniformly convex. Therefore, the previously known properties of XKW and the new geometric observation answer, in the negative, the question of Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov whether asymptotic midpoint uniform convexity and asymptotic uniform convexity are isomorphically equivalent and a question of Perreau whether asymptotic midpoint uniform convexity implies the Point of Continuity Property.

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Cite

BibTeXRIS
BibTeX
@misc{baudier2026kadetswerner,
  title = {{The Kadets--Werner modification of Bourgain--Rosenthal space is asymptotically midpoint uniformly convex}},
  author = {Florent P. Baudier},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21283},
  eprint = {2609.21283},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21283},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Kadets--Werner modification of Bourgain--Rosenthal space is asymptotically midpoint uniformly convex
AU  - Florent P. Baudier
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21283
UR  - https://arxiv.org/abs/2609.21283
AB  - Let X_KW be the Kadets--Werner modification of a closed subspace of L_1 constructed by Bourgain and Rosenthal in 1980. In this short note, it is shown that X_KW is asymptotically midpoint uniformly convex. Since X_KW has the Daugavet Property, it fails the Point of Continuity Property and thus does not admit an equivalent norm that is asymptotically uniformly convex. Therefore, the previously known properties of X_KW and the new geometric observation answer, in the negative, the question of Dilworth, Kutzarova, Randrianarivony, Revalski and Zhivkov whether asymptotic midpoint uniform convexity and asymptotic uniform convexity are isomorphically equivalent and a question of Perreau whether asymptotic midpoint uniform convexity implies the Point of Continuity Property.
ER  -