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The volume of a certain set of matrices

Published online by Cambridge University Press:  24 October 2008

Henry Jack  and
A. M. Macbeath
Henry Jack
Affiliation:
Queen's CollegeDundee
A. M. Macbeath
Affiliation:
Queen's CollegeDundee
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 55 , Issue 3 , July 1959 , pp. 213 - 223
Copyright
Copyright © Cambridge Philosophical Society 1959

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References

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