Refinement of the lower bound for the index in the octahedron problem (Q1908433)

In this paper the author gives a lower bound for the index in the octahedron problem. He proves that the maximum index \(I\) defined by the index of the sublattice \(\Lambda'\) of \(\Lambda\) spanned by a set of linearly independent vector systems giving an admissible octahedron of \(\Lambda\) is greater than \[ {n! \over 2^{n-1}} \bigl(1+(2+ \delta)^{-n} \bigr), \] for sufficiently large \(n\) (here \(\delta>0\) is an arbitrarily small number). The method of the proof uses an averaging procedure which naturally appears in such problems.