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arXiv · 2402.13869

A note on smooth rotund norms which are not midpoint locally uniformly rotund

Carlo Alberto De Bernardi·Alessandro Preti·Jacopo Somaglia
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Abstract

We prove that every separable infinite-dimensional Banach space admits a G\^ateaux smooth and rotund norm which is not midpoint locally uniformly rotund. Moreover, by using a similar technique, we provide in every infinite-dimensional Banach space with separable dual a Fr\'echet smooth and weakly uniformly rotund norm which is not midpoint locally uniformly rotund. These two results provide a positive answer to some open problems by A. J. Guirao, V. Montesinos, and V. Zizler.

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Carlo Alberto De Bernardi, Alessandro Preti, Jacopo Somaglia. 2024-02-21. A note on smooth rotund norms which are not midpoint locally uniformly rotund. https://arxiv.org/abs/2402.13869

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