The maximum of the smallest maximal coordinate of the minimum vectors in 6-lattices equals 1 (Q1770348)

Let \(\Lambda(A)\) be an \(n\)-lattice with basis \(A=\{a_i\}_{i=1}^n\), and having, at least, \(n\) linearly independent vectors all of them with length \(\lambda_1(\Lambda)\), which can be supposed equals 1 (minimum vectors); further more, let \[ L(\Lambda)=\min_{A\text{ basis of }\Lambda}\max_{| v| =\lambda_1(\Lambda)}\Bigl\{x_i\in{\mathbb Z}:v=\textstyle\sum_{i=1}^nx_ia_i\Bigr\}. \] Then \(L_n\) is defined as the maximum of the values \(L(\Lambda)\) when \(\Lambda\) ranges over all the \(n\)-lattices of the Euclidean space; thus, the problem arises to determine this number \(L_n\). In [Stud. Sci. Math. Hung. 29, No. 1--2, 169--175 (1994; Zbl 0808.52018)], \textit{Á. G. Horváth} proved some results regarding how to find a basis of a \(n\)-lattice such that the (absolute values of the) coordinates of these minimum vectors are ``small enough''; and moreover, he proved that in dimensions 4 and 5, \(L_4=L_5=1\), finding also the precise basis. In this paper, the author considers the case \(n=6\), and shows that also \(L_6=1\); that is, he proves that for any 6-lattice, there always exists a basis such that the maximum of the coordinates of a minimum vector is equal, at most, to 1.