Polyhedral renormings with property Δ and LFC structure

Determine whether every polyhedral Banach space admits an equivalent polyhedral norm with property (Δ), equivalently whether every polyhedral Banach space admits an equivalent polyhedral LFC norm; these conditions are also equivalent to admitting a locally finite tiling by bounded convex bodies.

Background

The paper proves that a normed space admits a locally finite tiling by bounded convex bodies if and only if it admits an equivalent (VI)-polyhedral norm with property (Δ), equivalently an equivalent (K)-polyhedral LFC norm. For Banach spaces, this yields the corresponding characterization in terms of polyhedral renormings with property (Δ).

The authors note that it remains unresolved whether every polyhedral Banach space has such a renorming. They further state that the existence of a polyhedral norm with property (Δ), the existence of a polyhedral LFC norm, and the existence of a locally finite tiling are equivalent questions in this setting.