Some inequalities related to the Seysen measure of a lattice

Given a lattice L, a basis B of L together with its dual B∗, the orthogonality measure of B was introduced by Seysen (1993) [9], . This measure (the Seysen measure in the sequel, also known as the Seysen metric [11]) is at the heart of the Seysen lattice reduction algorithm and is linked with different geometrical properties of the basis [6,7,10,11]. In this paper, we derive different expressions for this measure as well as new inequalities related to the Frobenius norm and the condition number of a matrix.