Optimal uniform constant for the complex rank-one spectral-norm problem

Determine the smallest constant c_0 that is valid uniformly over all dimensions n\geq 1 and all complex vectors a,b\in\mathbb{C}^n for the rank-one matrix G=a\mathbf{1}^{\mathsf T}-\mathbf{1}b^{\mathsf T} in the spectral-norm inequality \[ c_0\min_{W\ \mathrm{unitary}}\|W\circ G\|_2\geq \min_{P\ \mathrm{permutation}}\|P\circ G\|_2. \]

Background

For matrices of the form G=a\mathbf{1}{\mathsf T}-\mathbf{1}b{\mathsf T} with complex a,b, the paper identifies the problem of finding the best dimension-independent constant in the spectral-norm version of Problem 1. The constant must work simultaneously for every matrix dimension and every choice of complex vectors. The authors describe this optimization problem as a longstanding open problem and connect it to a problem in spectral matching for normal matrices.

References

For $G=a1{\T}-\boneb{\T}$ with $a,b\inCn$ and $|\cdot|_{\UI}=|\cdot|_2$, finding the smallest constant $c_0$ in eq:liconj-0 that is valid uniformly for all $n\ge1$ and all $a,b\inCn$ is a long standing open problem and is recently collected as Problem SP-07 of.

eq:liconj-0:

$c_0\cdot\min_{W\ \mathrm{unitary}}\|W\circ G\|_{\UI} \ge\min_{P\ \mathrm{permutation}}\|P\circ G\|_{\UI}. $

— On a Conjecture Related to Eigenvalue Perturbations  (2609.25552 - Zhang et al., 22 Sep 2026) in Section 5, Discussion