Normal tilings of a Banach space and its ball

Abstract

We show some new results about tilings in Banach spaces. A tiling of a Banach space Xis a covering by closed sets with non-empty interior, so that they have pairwise disjoint interiors. If,moreover, the tiles have inner radii uniformly bounded from below, and outer radii uniformly boundedfrom above, we say that the tiling is normal. In 2010, Preiss constructed a convex normal tiling ofthe separable Hilbert space. For Banach spaces with Schauder basis, we will show that Preiss’ resultis still true with starshaped tiles instead of convex ones. Also, whenever X is uniformly convex wegive precise constructions of convex normal tilings of the unit sphere, the unit ball or in general ofany convex body.