On a question by Corson about point-finite coverings

TL;DR

This paper affirms Corson's 1961 question by demonstrating that every Banach space can be covered with bounded convex tiles such that each point belongs to at most two tiles, providing a new tiling construction.

Contribution

It introduces a method to produce bounded convex tilings of Banach spaces with order 2, ensuring no point belongs to more than two tiles, solving a long-standing open problem.

Findings

Every Banach space admits a bounded convex tiling of order 2.

The tiling ensures disjoint interiors of tiles.

No point in the space belongs to more than two tiles.

Abstract

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 nonempty 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 nonempty interior (tiles) such that the interiors are pairwise disjoint and no point of X belongs to more than two tiles.