Vladimir P. Fonf,
e prove that, given any covering of any infinite-dimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a point-finite covering by the union of countably many slices of the unit ball.
point finite coverings, slices, polyhedral spaces, Hilbert spaces
46B20 - Geometry and structure of normed linear spaces
46C05 - Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
52C17 - Packing and covering in $n$ dimensions [See also 05B40, 11H31]