On isomorphically polyhedral $\mathcal L_\infty$-spaces

TL;DR

This paper demonstrates that certain $ ext{L}_ ext{infty}$-subspaces within separable isomorphically polyhedral Lindenstrauss spaces cannot be renormed to retain their Lindenstrauss space structure, revealing limitations in their geometric properties.

Contribution

It provides the first example of $ ext{L}_ ext{infty}$-subspaces that defy renorming into Lindenstrauss spaces, highlighting new geometric constraints.

Findings

Existence of $ ext{L}_ ext{infty}$-subspaces that cannot be renormed as Lindenstrauss spaces

Limitations on renorming properties of isomorphically polyhedral spaces

New insights into the structure of Lindenstrauss and $ ext{L}_ ext{infty}$-spaces

Abstract

We show that there exist $\mathcal L_\infty$-subspaces of separable isomorphically polyhedral Lindenstrauss spaces that cannot be renormed to be a Lindenstrauss space.