Octahedral norms and convex combination of slices in Banach spaces
By J. Guerrero, G. L'opez-P'erez, Abraham Rueda Zoca. 2013
67 citations recorded by the source corpus.
We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach spaces. We prove that the norm of an arbitrary Banach space is octahedral if, and only if, every convex combination of $w^*$-slices in the dual unit ball has diameter 2, which answer an open question. As a consequence we get that the Banach spaces with the Daugavet property and its dual spaces have octahedral norms. Also, we show that for every separable Banach space containing $\ell_1$ and for every $\varepsilon >0$ there is an equivalent norm so that every convex combination of $w^*$-slices in the dual unit ball has diameter at least $2-\varepsilon$.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/15a4617e460085bf01d395eb91869979f9419444
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In Computational Theory and Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- On strongly norm attaining Lipschitz maps (2018, result): “From an isometric point of view, it is proved in [14, Theorem 1 and Proposition 3] that if a Banach space X has an octahedral norm, then every convex combination of weak-star slices of BX∗ has diameter two, and the reciprocal result has been recently proved in [5].”