Tilings and coverings by balls in ℓ1
Carlo Alberto De Bernardi, Tommaso Russo, Şeyda Sezgek, Jacopo Somaglia
Source abstract
Abstract A famous result of Klee from 1981 is that the Banach space admits a disjoint tiling by balls of radius 1, for all cardinals with . Klee also observed that the smallest cardinal in which such a tiling might exist is , leaving open the question whether, for , might admit a tiling by balls at all. Our main result answers this question in the negative, proving in particular that does not admit any tiling by balls. We also give a companion result about coverings by balls of and we give a construction of a star‐finite tiling of , for each space whose dimension is at most countable.
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