A general framework for extending means to higher orders

Although there is an extensive literature on various means of two positive operators and their applications, these means do not typically readily extend to means of three and more operators. It has been an open problem to define and prove the existence of these higher order means in a general setting. In this paper we lay the foundations for such a theory by showing how higher order means can be inductively defined and established in general metric spaces, in particular, in convex metric spaces. We consider uniqueness properties and preservation properties of these extensions, properties which provide validation to our approach. As our targeted application, we consider the positive operators on a Hilbert space under the Thompson metric and apply our methods to derive higher order extensions of a variety of standard operator means such as the geometric mean, the Gauss mean, and the logarithmic mean. That the operator logarithmic mean admits extensions of all higher orders provides a positive solution to a problem of Petz and Temesi [SIAM J. Matrix Anal. Appl. 27 (2005)].