  Let A be an n by n matrix with determinant 1. We show that for all n > 2
there exist dimensional strictly positive constants C_n such that the average
over the orthogonal group of log rho(A X) d X > C_n log ||A||, where ||A||
denotes the operator norm of A (which equals the largest singular value of A),
rho denotes the spectral radius, and the integral is with respect to the Haar
measure on O_n The same result (with essentially the same proof) holds for the
unitary group U_n in place of the orthogonal group. The result does not hold in
dimension 2. We also give a simple proof that the average value over the unit
sphere of log ||A u|| is nonnegative, and vanishes only when A is orthogonal.
