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Tilings of Normed Spaces

Published online by Cambridge University Press:  20 November 2018

Carlo Alberto De Bernardi  and
Libor Veselý
Carlo Alberto De Bernardi
Affiliation:
Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano e-mail: carloalberto.debernardi@gmail.com, libor.vesely@unimi.it
Libor Veselý
Affiliation:
Dipartimento di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano e-mail: carloalberto.debernardi@gmail.com, libor.vesely@unimi.it
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Abstract

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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 was 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 the 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.

  • (i) $X$ admits no tiling by Fréchet smooth bounded tiles.

  • (ii) If $X$ is locally uniformly rotund $\left( \text{LUR} \right)$, it does not admit any tiling by balls.

  • (iii) On the other hand, some ${{\ell }_{1}}\left( \Gamma \right)$ spaces, $\Gamma $ uncountable, do admit a tiling by pairwise disjoint $\text{LUR}$ bounded tiles.

Keywords

46B2052A9946A45tiling of normed spaceFréchet smooth bodylocally uniformly rotund bodyl1(Γ)- spaces
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 2 , 01 April 2017 , pp. 321 - 337
Copyright
Copyright © Canadian Mathematical Society 2017

References

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