Some Inequalities Related to the Seysen Measure of a Lattice

Given a lattice $L$, a basis $B$ of $L$ together with its dual $B^{\ast }$, the orthogonality measure $S(B)={\sum }_i\mathrm{\mid }\mathrm{\mid }{b}_i\mathrm{\mid }{\mathrm{\mid }}^2\mathrm{\mid }\mathrm{\mid }{b}_i^{\ast }\mathrm{\mid }{\mathrm{\mid }}^2$ of $B$ was introduced by M. Seysen in 1993. This measure is at the heart of the Seysen lattice reduction algorithm and is linked with different geometrical properties of the basis. In this paper, we explicit different expressions for this measure as well as new inequalities.