(IV)-polyhedral renormings for separable polyhedral normed spaces

Determine whether every separable polyhedral normed space admits an equivalent (IV)-polyhedral renorming.

Background

The paper recalls that every separable polyhedral Banach space admits an (IV)-polyhedral renorming. It then establishes a related result for separable (VI)-polyhedral normed spaces with property (Δ), showing that these spaces admit an (IV)-polyhedral renorming together with smooth and analytic approximations.

The authors explicitly leave unresolved whether the same renorming conclusion holds for all separable polyhedral normed spaces, without assuming completeness or the additional (VI) and (Δ) hypotheses.

References

While we don't know if this is true for separable polyhedral normed spaces, we can prove it for (VI)-polyhedral normed spaces with ($\Delta$).

— Polyhedral normed spaces: the structural theorem and locally finite tilings  (2609.29279 - Bernardi et al., 24 Sep 2026) in Section 3, The structural theorem, paragraph following Corollary 3.10