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# Tilings of Normed Spaces

Published:2016-03-08
Printed: Apr 2017
• Carlo Alberto De Bernardi,
Dipartimento di Matematica , Università degli Studi , Via C. Saldini 50 , 20133 Milano , Italy
• Libor Veselý,
Dipartimento di Matematica , Università degli Studi , Via C. Saldini 50 , 20133 Milano , Italy
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## Abstract

By a tiling of a topological linear space $X$ we mean a covering of $X$ by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite-dimensional spaces initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space $X$, our main results are the following. 1. $X$ admits no tiling by Fréchet smooth bounded tiles. 2. If $X$ is locally uniformly rotund (LUR), it does not admit any tiling by balls. 3. On the other hand, some $\ell_1(\Gamma)$ spaces, $\Gamma$ uncountable, do admit a tiling by pairwise disjoint LUR bounded tiles.
 Keywords: tiling of normed space, Fréchet smooth body, locally uniformly rotund body, $\ell_1(\Gamma)$-space
 MSC Classifications: 46B20 - Geometry and structure of normed linear spaces 52A99 - None of the above, but in this section 46A45 - Sequence spaces (including Kothe sequence spaces) [See also 46B45]

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