A proof of a conjecture of C.L. Siegel

Let M_n be the Minkowski fundamental domain for the space of n × n real, symmetric, and positive definite matrices under the action of the unimodular group SL_n(Z). C. L. Siegel conjectured that d(A, B) - f(A, B) ≤ C(n), for A, B ∈ M_n, where d and f are the geodesic and the reduced geodesic distances, respectively, and C(n) is a constant depending only on n. This conjecture appears in his book ("Zur Reduktionstheorie Quadratischer Formen," The Mathematical Society of Japan, 1959). By reducing the problem to the diagonal matrices in M_n, we obtain a proof.