Polyhedral normed spaces: the structural theorem and locally finite tilings

Abstract

A fundamental result due to Fonf (1981) asserts that the unit sphere of every polyhedral Banach space is covered by true faces of the unit ball. The principal aim of our paper is to study the validity of the same result for polyhedral normed spaces and present some applications. Our first main result is that if $X$ is a polyhedral normed space with property ($Δ$), then its unit sphere is covered by algebraic true faces. It then follows that if the space is additionally (VI)-polyhedral, then its unit sphere is covered by genuine true faces. We also present several counterexamples showing that these results are optimal in a strong sense. For instance, we show that there exist (V)-polyhedral normed spaces whose unit ball doesn't have any algebraic true face at all, and that there exist polyhedral normed spaces with ($Δ$) whose unit sphere is not covered by true faces. Further, we give an example of a polyhedral normed space such that the set of strongly exposed points of the unit ball is dense in the sphere, and some results and counterexamples concerning boundaries of polyhedral normed spaces. Our principal application of this material involves locally finite tilings of normed spaces, and we show that a normed space admits a locally finite tiling (by bounded convex bodies) if and only if it admits a (VI)-polyhedral norm with ($Δ$). In particular, such a tiling exists in every polyhedral Banach space with ($Δ$), which generalises a result of Fonf.