Type: Article

Almost disjoint families and the geometry of nonseparable spheres

Guzmán, Osvaldo; Hrušák, Michael; Koszmider, Piotr;

Abstract:

We consider uncountable almost disjoint families of subsets of , the Johnson-Lindenstrauss Banach spaces induced by them, and their natural equivalent renormings . We introduce a partial order and characterize some geometric properties of the spheres of and of in terms of combinatorial properties of . This provides a range of independence and absolute results concerning the above mentioned geometric properties via combinatorial properties of almost disjoint families. Exploiting the extreme behavior of some known and some new almost disjoint families 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 there is a nonseparable Banach space, where for every there is such that every uncountable -separated set of elements of the unit sphere contains two elements distant by less than 1 and two elements distant at least by by . It should be noted that for every every nonseparable Banach space has a plenty of uncountable -separated sets by the Riesz Lemma. We also obtain a consistent dichotomy for the spaces of the form : The Open Coloring Axiom implies that the unit sphere of every Banach space of the form either is the union of countably many sets of diameter strictly less than 1 or it contains an uncountable -separated set for every