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 spaceswas 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 (LUR),
it does not admit any tiling by
balls.
(iii)
On the other hand, some spaces, г
uncountable, do admit a tiling by pairwise
disjoint LUR bounded tiles.