Asymptotic Growth of a Matrix Cone Norm Constant

Definitions and mathematical statement

Let $n\in{\mathbb{N}}$, $n\geq1$, and let ${\mathbb{C}}^{n\times n}$ denote the complex matrices of size $n\times n$, reading the supplied matrix domain as the complex numbers. For such a matrix, $A\succeq0$ means that $A=A^*$ and $z^*Az\geq0$ for every $z\in{\mathbb{C}}^n$, where $A^*$ is the conjugate transpose.

The supplied problem asks for the asymptotic growth, as $n\to\infty$, of \[ C_n=\sup\left\{\frac{\gamma_+(A)}{\lVert A\rVert_{(e,1)}}:A\in{\mathbb{C}}^{n\times n},\ A\succeq0,\ A\neq0\right\}. \] However, the supplied record does not define either $\gamma_+(A)$ or $\lVert A\rVert_{(e,1)}$. These symbols are retained as source notation; their precise definitions and normalizations must be recovered from the cited sources before this formula specifies a fully defined sequence. No particular factorization functional or matrix norm is assumed here.

The supplied report excerpt attributes a lower bound $C_n\geq c\sqrt n$, with a universal constant $c>0$, to work by Mixon, Steinerberger, and the excerpt's unidentified first-person speaker, referring to its reference [6]. It also reports an upper bound of order $n$. The supplied dated curation describes these bounds as still leaving the asymptotic growth undetermined. These are supplied source claims, not independently verified bounds or a new literature assessment.

Short English statement

Determine how $C_n$, the supremum of the displayed ratio over nonzero complex positive semidefinite matrices of size $n\times n$, grows as $n\to\infty$. The supplied record labels the problem open and reports bounds between constant multiples of $\sqrt n$ and $n$. A complete formulation requires the missing definitions of $\gamma_+$ and $\lVert\cdot\rVert_{(e,1)}$.

Sources

• Supplied local registry record: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30005892$, source problem code catalogue entry, title Asymptotic Growth of a Matrix Cone Norm Constant. The registry supplies the status open and the statement status corrected_verified, together with a claim of comparison against the official report; these metadata have not been independently verified here.
• Applied Harmonic Analysis and Data Science (2024), Oberwolfach Reports, pp. 1163--1226, DOI 10.4171/owr/2024/21: https://doi.org/10.4171/owr/2024/21. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled Bandeira--Mixon--Steinerberger: https://arxiv.org/abs/2405.06154. The excerpt refers to reference [6], but does not supply its bibliographic entry or explicitly identify the first-person speaker.
• Supplied literature source labeled OWR report: https://ems.press/content/serial-article-files/49476.
• Supplied literature source labeled Balan--Jiang factorization: https://arxiv.org/abs/2409.20372.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the best cited bounds remain a constant times $\sqrt n$ below and order $n$ above. This is supplied dated curation derived from the OWR material, not an independently verified assessment. Its time-sensitive status and cited claims require independent checking.