Strongly regular Banach spaces with big weakly open subsets in the unit ball

Abstract

We construct, given $1<p<\infty$, a Banach space $Y$ and a closed, convex and symmetric set $L\subseteq B_Y$ with the following properties: 1) $Y^{\ast \ast }$ is strongly regular (henceforth, $Y$ is strongly regular). 2) Every non-empty relatively weakly-star open subset of ${\overline{L}}^{w^{\ast }}$ (the $w^{\ast }$ closure of $L$ in $Y^{\ast \ast }$) has radius one. In particular, every non-empty relatively weakly open subset of $L$ has radius $1$. 3) Every non-empty relatively weakly open subset of $L$ has diameter, at least, $2^{\frac{1}{p}}$. This constitutes an advance to the question whether there exists a strongly regular Banach spaces satisfying that every non-empty relatively weakly open subset of the unit ball has radius $1$. As a partial answer, we get that for every $\epsilon >0$ there exists a strongly regular Banach spaces where weakly open subsets have radius, at least, $1-\epsilon$.

26 pages

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