A strictly convex and smooth Banach space with the Daugavet property
Sheldon Dantas, Jaan Kristjan Kaasik, Yoël Perreau
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.
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