Printed: Apr 2017
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
Study of tilings of infinite-dimensional spaces initiated in
1980's with pioneer papers by V. Klee.
We prove some general properties of tilings of locally convex
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
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.
tiling of normed space, Fréchet smooth body, locally uniformly rotund body, $\ell_1(\Gamma)$-space
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]