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A strictly convex and smooth Banach space with the Daugavet property
Abstract
We construct equivalent norms on L1[0,1], arbitrarily close to the usual L1-norm, which are weakly midpoint locally uniformly rotund and Gâteaux smooth, and whose dual norms have the Daugavet property. In particular, we obtain a strictly convex and smooth Banach space with the Daugavet property, thereby solving a longstanding open problem in the area.
Related
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BibTeX
@misc{dantas2026strictly,
title = {{A strictly convex and smooth Banach space with the Daugavet property}},
author = {Sheldon Dantas and Jaan Kristjan Kaasik and Yoël Perreau},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12562},
eprint = {2609.12562},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12562},
}
RIS
TY - UNPB
TI - A strictly convex and smooth Banach space with the Daugavet property
AU - Sheldon Dantas
AU - Jaan Kristjan Kaasik
AU - Yoël Perreau
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12562
UR - https://arxiv.org/abs/2609.12562
AB - We construct equivalent norms on L_1[0,1], arbitrarily close to the usual L_1-norm, which are weakly midpoint locally uniformly rotund and Gâteaux smooth, and whose dual norms have the Daugavet property. In particular, we obtain a strictly convex and smooth Banach space with the Daugavet property, thereby solving a longstanding open problem in the area.
ER -