The defects of admissible balls and octahedra in a lattice, and systems of generic representatives
Andrei Mikhailovich Raigorodskii · Sbornik Mathematics · 1998
Let be the frame of unit coordinate vectors, let such that , let be the unit octahedron, and let be the unit ball. A set is said to be admissible in if . The defect , with respect to , of a set admissible in is the smallest number of vectors to be deleted from in order that the remaining system can be complemented to a basis in . Let and let , where the maximum is taken over all in the first case and over all such that is a cyclic group in the second. It is shown that and , where is an absolute constant.These results are obtained using methods of geometry and combinatorial analysis.