Abstract view
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
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.