On two questions concerning tilings (Q1261910)

The paper investigates a question by \textit{V. Klee} [Stud. Sci. Math. Hung. 21, 415-427 (1987; Zbl 0577.52007)] if there is a tiling of separable Hilbert space by bounded convex tiles. The author proves the following Theorem. Let \(T\) be a convex tiling of a Banach space \(X\) and let \(Y\subset X\) be an infinite-dimensional reflexive separable closed subspace. If \(C_ 0\in T\) is bounded and if \(x\in\text{int} C_ 0\), then the flat \(Y+x\) contains uncountably many points of \(bd C_ 0\backslash{\mathcal P}(T)\). Here \({\mathcal P}(T)\) denotes the set of frontier points of \(T\) which are interior to the union of the tiles that contain it. Moreover the paper confirms the conjecture of \textit{A. Valette} [Bull. Soc. Math. Belg., Ser. A 32, 11-19 (1980; Zbl 0486.05023)] that a tiling of a plane by topological disks is locally finite at most boundary points of tiles.