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# Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls

Published:2012-09-21
Printed: Mar 2014
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.
 MSC Classifications: 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]