Randomstrasse101 Open Problems (2025) — Conjecture 1 (Type-$2$ constant of Tensors)

Definitions and mathematical statement

The supplied record presents the following as Conjecture 1, concerning the type-$2$ constant of tensors. For $p\geq 2$, it asserts \[ \mathbb{E}\left\|\sum_{i=1}^{n}g_iT_i\right\|_{\mathcal{I}_p} \leq \widetilde{\mathcal{O}}_{r,p}\left( d^{\frac12-\frac1p} \left(\sum_{i=1}^{n}\|T_i\|_{\mathcal{I}_p}^{2}\right)^{1/2} \right). \] Here the supplied meaning of $\widetilde{\mathcal{O}}_{r,p}$ is an upper bound allowing multiplicative constants depending on $r$ and $p$, together with polylogarithmic factors in $d$ and $n$.

The supplied excerpt does not define the tensor space, the scalar field, the quantity $\|\cdot\|_{\mathcal{I}_p}$, or the roles and admissible values of $r$ and $d$. The index $n$ counts the displayed summands, but the admissible families $(T_i)_{i=1}^{n}$ are not specified. Neither the joint probability law of $(g_i)_{i=1}^{n}$ nor the probability space underlying $\mathbb{E}$ is given. In particular, Gaussianity, independence, normalization, and whether the tensors are deterministic cannot be assumed from this excerpt. The admissible range of $p$ beyond the written condition $p\geq 2$, including whether an endpoint $p=\infty$ is intended, is also unspecified. Thus the displayed assertion preserves the source formula, but its full quantified formulation remains unresolved.

Short English statement

The supplied conjecture bounds the expected $\mathcal{I}_p$ size of a weighted tensor sum by $d^{1/2-1/p}$ times the square root of the sum of squared individual sizes, allowing constants depending on $r,p$ and polylogarithmic factors in $d,n$. Definitions of the tensors, their size, and the random coefficients are missing, so a complete mathematical statement requires review.

Sources

• Local UnsolvedMath source registry, permanent ID $800861$, source problem code catalogue entry. The record attributes the problem to Bandeira et al., gives proposed year $2026$, and labels its status as open. This status is supplied registry data, not an independently verified mathematical assessment.
• Bandeira et al., Randomstrasse101: Open Problems of 2025 ($2026$), Conjecture 1, context: type-$2$ constant of tensors. Supplied source URL: https://arxiv.org/abs/2603.29571. The record reports extraction by arxiv_latex and an access date of 2026-10-16; these are supplied provenance details, not a claim of independent source access or verification.