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 $$ for $$ 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 $$.