Geodesic in-betweenness for means of several matrices

A matrix mean σ is said to satisfy the in-betweenness property with respect to the metric d if for any pair of positive definite matrices A and B,d(A,AσB)≤d(A,B). In this article, we introduce the geodesic in-betweenness property for k-tuples of positive definite matrices with respect to any metric. Moreover, we show that in spaces of non-positive curvature, geodesic in-betweenness implies in-betweenness in the case of two matrices. We then show some examples of metrics and means for which this property is satisfied.