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arXiv · 2609.12562

A strictly convex and smooth Banach space with the Daugavet property

Sheldon Dantas·Jaan Kristjan Kaasik·Yoël Perreau
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Abstract

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.

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Sheldon Dantas, Jaan Kristjan Kaasik, Yoël Perreau. 2026-09-11. A strictly convex and smooth Banach space with the Daugavet property. https://arxiv.org/abs/2609.12562

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