Plasticity of unit balls of Banach spaces (single-space and two-space versions)

Determine whether, for arbitrary Banach spaces X and Y (including the special case X = Y), every non-expansive bijection between the unit balls B_X and B_Y is an isometry; equivalently, establish whether the unit ball of an arbitrary Banach space is plastic and whether this property extends to pairs of (possibly different) Banach spaces.

Background

A substantial body of work studies when non-expansive bijections between unit balls of Banach spaces must be isometries, with many positive results in particular settings and under additional assumptions. Nonetheless, a general resolution for arbitrary Banach spaces remains elusive.

The authors highlight that, despite progress (including conditional results and examples in special classes), there are no known counterexamples of Banach spaces with non-plastic unit balls. The question persists both for a single space (self-bijections of its unit ball) and for bijections between unit balls of two possibly different Banach spaces.

References

However, the problem of plasticity for the unit ball of an arbitrary Banach space is still open as well as its extension to the case of two different spaces.

— Plastic metric spaces and groups  (2510.10537 - Banakh et al., 12 Oct 2025) in Introduction

Whether the unit ball of every Banach space is plastic remains open, although affirmative answers are known for several classes, including $\ell_1$-sums of strictly convex spaces , $c$ and $c_0$ , certain $C(K)$ spaces , and the real space $\ell_\infty$ .

— Plasticity in graph metric spaces and their hyperspaces  (2609.39593 - Hida, 30 Sep 2026) in Section 1, Introduction