Octahedral norms and convex combination of slices in Banach spaces

TL;DR

This paper establishes a precise equivalence between octahedral norms and the maximal diameter of convex combinations of slices in Banach spaces, providing new insights into the geometric structure of these spaces.

Contribution

It proves that a Banach space has an octahedral norm if and only if all convex combinations of $w^*$-slices in its dual have diameter 2, answering an open question.

Findings

Banach space norm is octahedral iff convex combinations of $w^*$-slices in dual have diameter 2

Spaces with the Daugavet property have octahedral norms

Existence of equivalent norms making convex combinations of slices nearly maximal in diameter

Abstract

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 $2-\varepsilon$.