Symmetry of Birkhoff–James orthogonality under the S0I property

Prove that every normed space with the simultaneous zero inclusion property has symmetric Birkhoff–James orthogonality; that is, establish that x perpendicular to y in the Birkhoff–James sense implies y perpendicular to x for all vectors x and y.

Background

The simultaneous zero inclusion (S0I) property requires that, for every invertible bounded linear operator T, inclusion of zero in the spatial numerical range W(T) implies inclusion of zero in W(T{-1}). The paper proves that, for several broad classes of Banach spaces—including smooth and almost locally uniformly rotund spaces, finite-dimensional smooth or strictly convex spaces, finite-dimensional polyhedral spaces, and two-dimensional real Banach spaces—the S0I property forces symmetry of Birkhoff–James orthogonality.

The authors state that they were unable to find any Banach space possessing the S0I property while having nonsymmetric Birkhoff–James orthogonality. They therefore formulate the stronger conjecture that the implication holds for every normed space, without smoothness, rotundity, finite-dimensionality, or other geometric assumptions. The conjecture remains unresolved in the paper.

References

Throughout our investigation, we are unable to find any Banach space with the $S0I$ property for which Birkhoff-James orthogonality fails to be symmetric. This observation naturally leads us to the following stronger conjecture: Suppose that $X$ is a normed space with the $S0I$ property. Then Birkhoff-James orthogonality in $X$ is symmetric. Whether this conjecture holds in full generality remains an intriguing question and, in our view, deserves further investigation.

— On the simultaneous zero inclusion property and characterization of Inner product space  (2610.13378 - Ghosh, 9 Oct 2026) in Concluding remark