On a question by Corson about point-finite coverings

We answer in the affirmative the following question raised by H. H. Corson in 1961: “Is it possible to cover every Banach space X by bounded convex sets with non-empty interior in such a way that no point of X belongs to infinitely many of them?” Actually, we show the way to produce in every Banach space X a bounded convex tiling of order 2, i.e., a covering of X by bounded convex closed sets with non-empty interior (tiles) such that the interiors are pairwise disjoint and no point of X belongs to more than two tiles.