10 Lectures and 42 Open Problems — Lata $\l$ a-Riemer-Schutt

Definitions and mathematical statement

Let $d\geq1$, let $e_1,\ldots,e_d$ be the standard basis of ${\mathbb{R}}^d$, and let $\|X\|=\sup_{\|v\|_2=1}\|Xv\|_2$ be the Euclidean operator norm. The source asks for a universal comparison \[ \mathbb E\|X\|\leq C\,\mathbb E\max_{1\leq k\leq d}\|Xe_k\|_2, \] where $C<\infty$ is independent of $d$ and of the entry variances, and $X$ is a real symmetric Gaussian matrix whose upper-triangular entries are independent. The expectation remains outside the column maximum.

The intended centered arbitrary-variance model is \[ X_{ij}=b_{ij}g_{ij}\quad(i\leq j),\qquad X_{ji}=X_{ij}, \] where $b_{ij}\geq0$ are arbitrary deterministic numbers and the $g_{ij}$, $i\leq j$, are independent standard real Gaussians. Zero variances are permitted in this model. Centering is not explicitly stated in the extracted question, and is an essential missing qualification: arbitrary deterministic means can yield matrices with operator norm much larger than their maximum column norm as dimension grows. Thus the centered model is identified as the intended conjectural setting, not silently claimed to be present in the source wording.

The reverse comparison holds pointwise, $\max_k\|Xe_k\|_2\leq\|X\|$, so the requested upper bound gives equivalence up to a universal constant. The supplied research summary reports this comparison and its reverse settled for centered arbitrary-variance symmetric Gaussian matrices by Latała--van Handel--Youssef (2018). No explicit universal constant or independent proof is supplied here.

Short English statement

The question compares the expected norm of a symmetric Gaussian matrix with the expected largest column norm, uniformly over dimension and variances. The supplied summary reports the centered Gaussian conjecture solved. The original extraction omits centering, which is necessary for the universal statement.

Sources

• Local UnsolvedMath registry, permanent ID 2800402, exact source code catalogue entry, Open Problem 4.2; status solved, classification SOLVED-IN-LITERATURE. The record names Afonso S. Bandeira as proposer in 2015.
• Bandeira, 42 Open Problems in Data Science (source-list year 2016), Open Problem 4.2, author blog https://afonsobandeira.wordpress.com/2015/10/25/10l42concentration/. The provenance records HTML extraction, accessed July 29, 2026, and says the source presented the problem as open. The registry's item title associates it with Latała--Riemer--Schutt, with malformed accent markup in the supplied title.
• Latała--van Handel--Youssef (2018), cited by the supplied research summary for the arbitrary-variance symmetric Gaussian comparison. No title, publication details, or theorem number is supplied.