The dominant method for defining multivariate operator means is to express
them as fix-points under a contraction with respect to the Thompson metric.
Although this method is powerful, it crucially depends on monotonicity. We are
developing a technique to prove the existence of multivariate operator means
that are not necessarily monotone. This gives rise to an entire new class of
non-monotonic multivariate operator means.Comment: We discovered that an argument in the proof of the last part of the
last theorem in the first version of the paper is plainly wrong. Based on
examples we still have ground to believe that the theorem is correct, but
this is now only a conjecture. Some mean inequalities are adde
