Tilings and coverings by balls in ℓ1

A famous result of Klee from 1981 is that the Banach space & lscr;(1) (kappa) admits a disjoint tiling by balls of radius 1, for all cardinals kappa with kappa N-0 = kappa. Klee also observed that the smallest cardinal in which such a tiling might exist is kappa = 2 (N)(0) , leaving open the question whether, for kappa < 2(0)(N) , & lscr;(1) (kappa) might admit a tiling by balls at all. Our main result answers this question in the negative, proving in particular that & lscr;(1) does not admit any tiling by balls. We also give a companion result about star-n-finite coverings by balls of & lscr;(1) (kappa) and we give a construction of a star-finite tiling of chi circle plus(infinity) c(00), for each space chi whose dimension is at most countable.