Almost disjoint families and the geometry of nonseparable spheres

Abstract

We consider uncountable almost disjoint families of subsets of $\mathbb N$, the Johnson-Lindenstrauss Banach spaces $(\mathcal X_{\mathcal A}, \|\ \|_\infty)$ induced by them, and their natural equivalent renormings $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$. We introduce a partial order $\mathbb P_{\mathcal A}$ and characterize some geometric properties of the spheres of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty})$ and of $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$ in terms of combinatorial properties of $\mathbb P_{\mathcal A}$. Exploiting the extreme behavior of some known and some new almost disjoint families among others we show the existence of Banach spaces where the unit spheres display surprising geometry: 1) There is a Banach space of density continuum whose unit sphere is the union of countably many sets of diameters strictly less than $1$. 2) It is consistent that for every $\rho>0$ there is a nonseparable Banach space, where for every $\delta>0$ there is $\varepsilon>0$ such that every uncountable $(1-\varepsilon)$-separated set of elements of the unit sphere contains two elements distant by less than $1$ and two elements distant at least by $2-\rho-\delta$. It should be noted that for every $\varepsilon>0$ every nonseparable Banach space has a plenty of uncountable $(1-\varepsilon)$-separated sets by the Riesz Lemma. We also obtain a consistent dichotomy for the spaces of the form $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$: The Open Coloring Axiom implies that the unit sphere of every Banach space of the form $(\mathcal X_{\mathcal A}, \|\ \|_{\infty, 2})$ either is the union of countably many sets of diameter strictly less than $1$ or it contains an uncountable $(2-\varepsilon)$-separated set for every $\varepsilon>0$.