Singular points for tilings of normed spaces

A point x in a normed space X is said to be singular for a given tiling of X whenever each neighborhood of x intersects infinitely many tiles. We show that, when X is infinite-dimensional and all tiles are convex, special points in the boundary of tiles (like extreme points or PC points, if any) must be singular. Under the further assumptions that X is separable and doesn’t contain c₀, singular points abound among the smooth points of any bounded tile. Finally, in any normed space a tiling is constructed which is free of singular points and whose members are both bounded and star-shaped; this disproves the conjecture that Corson’s theorem might apply to star-shaped bounded coverings.