BANACH SPACES IN WHICH LARGE SUBSETS OF SPHERES CONCENTRATE
BANACH SPACES IN WHICH LARGE SUBSETS OF SPHERES CONCENTRATE
Abstract We construct a nonseparable Banach space $\mathcal {X}$ (actually, of density continuum) such that any uncountable subset $\mathcal {Y}$ of the unit sphere of $\mathcal {X}$ contains uncountably many points distant by less than $1$ (in fact, by less then $1-\varepsilon $ for some $\varepsilon>0$ ). This solves in …