Determination of means by invariance

If M is a mean on Rn + and M(f (x1), f (x2), . . . , f (xn)) = f (M(x1, x2, . . . , xn)) then we say that M is invariant under f . The problem is to find a class of functions that by invariance determines a mean uniquely. We focus on the geometric mean, which can be transformed to obtain results for other means. © 2005 Elsevier Inc. All rights reserved