Paper 2022/677

A Conjecture on Hermite Constants

Abstract

As of today, the Hermite constants $\gamma_n$ are only known for $n\in \{1,2,3,4,5,6,7,8,24\}$. We noted that the known values of $(4/\gamma_n)^n$ coincide with the values of the minimal determinants of any $n$-dimensional integral lattice when the length of the smallest lattice element $\mu$ is fixed to 4. Based on this observation, we conjecture that the values of $\gamma_n^n$ for $n=9,\ldots,23$ are those given in Table 2. We provide a supporting argument to back this conjecture. We also provide a provable lower bound on the Hermite constants for $1\leq n\leq24$.