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<p>We construct a Banach space <i>X</i> with the r-BSP such that the infimum of the diameter of the slices of the unit ball is 1, which gives negative answer to a 2006 question by Y. Ivakhno in an extreme way. This example is performed by considering modifications of the classical James-tree space <i>JT</i><sub>∞</sub> constructed on a tree with infinitely many branching points <i>T</i><sub>∞</sub>. Moreover, we prove that every Banach space with the Daugavet property admits, for every <i>ε</i> &gt; 0, an equivalent renorming for which its Daugavet index of thickness is bigger than 2 − <i>ε</i> and there are slices of the unit ball of diameter strictly smaller than 2, which solves an open question from [7].</p>

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Diameter, radius and Daugavet index of thickness of slices in Banach spaces

  • 期刊论文

  • 发表日期：

    2025年12月11日

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  • Abraham Rueda Zoca

摘要

We construct a Banach space X with the r-BSP such that the infimum of the diameter of the slices of the unit ball is 1, which gives negative answer to a 2006 question by Y. Ivakhno in an extreme way. This example is performed by considering modifications of the classical James-tree space JT∞ constructed on a tree with infinitely many branching points T∞. Moreover, we prove that every Banach space with the Daugavet property admits, for every ε > 0, an equivalent renorming for which its Daugavet index of thickness is bigger than 2 − ε and there are slices of the unit ball of diameter strictly smaller than 2, which solves an open question from [7].