Locally countable plump tilings are flat (Q1078812)

A (convex) body is said to be plump if its boundary is flat wherever it is conical. All smooth bodies and all rotund bodies are plump. It is proved that if a topological linear space X is such that each closed subset of X is a Baire space (thus, in particular, if X is a complete metric linear space), if \({\mathcal C}\) is a locally countable packing of plump bodies in X, and if v is a nonflat point of a member of \({\mathcal C}\), then arbitrarily close to v are points of X not covered by \({\mathcal C}\). From this it follows that the only locally countable tilings of X by plump convex bodies are those that consist of parallel strips. In particular, there is no tiling of the plane by bounded smooth convex bodies, thus settling a problem raised earlier. However, it is shown that the plane can be covered by nonoverlapping topological 2-balls each of which has a smooth boundary and whose diameters are uniformly bounded.