The values of the Hermite constants for dimensions other than 1–8 and 24

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $d\geq1$, a full-rank lattice is $L=B\mathbb Z^d\subset\mathbb R^d$ for an invertible real $d\times d$ matrix $B$. Let $\operatorname{covol}(L)=|\det B|$ and $\lambda_1(L)=\min\{\|v\|:v\in L\setminus\{0\}\}$, using Euclidean length. The Hermite constant is \[\gamma_d=\sup_L\frac{\lambda_1(L)^2}{\operatorname{covol}(L)^{2/d}}.\] The historically listed question concerns dimensions other than $1,\ldots,8$ and $24$. The cited 2026 version reports $\gamma_9=2$, addressing the nine-dimensional instance of that historical request.

Statement recorded in the supplied source.

Determine the exact values of $\gamma_d$ in the dimensions $d\notin\{1,2,\ldots,8,24\}$, with the source-supported value $\gamma_9=2$ already available. [S2]

Short English statement

Determine the exact Hermite constants outside dimensions one through eight and twenty-four. The cited version supplies the value two in dimension nine.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7200067 (AMR-071-0067). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mathieu Dutour Sikirić and Wessel van Woerden, The lattice packing problem in dimension 9 by Voronoi’s algorithm (arXiv:2508.20719v2, 2026). Section 1, Corollary 1, and Section 2.1, pp. 1–4. https://arxiv.org/pdf/2508.20719v2.