The volume of a certain set of matrices

Abstract

The object of the present paper is to evaluate m ( k ), the invariant measure (defined in §2) of the set of real matrices with determinant 1 and norm bounded by a positive real number k . The norm is that induced on the linear transformations of Euclidean n -space by the usual Euclidean norm in that space, and is also defined in § 2. We were led to this problem as a result of work arising from the Geometry of Numbers. A knowledge of the behaviour of m ( k ) was necessary in order to generalize a certain mean value formula (Macbeath and Rogers (9)) from Riemann to Lebesgue integrable functions.