The Karcher mean of equilateral triangles

In this paper we present a class of Riemannian equilateral triangles ΔABC of the Cartan-Hadamard Riemannian manifold of N×N positive definite matrices of determinant 1 whose Karcher mean of vertices is of the formΛ(A,B,C)=A+B+Cdet⁡(A+B+C)N. The obtained class of triples (A,B,C) forms an analytic manifold with an analytic parameterization over SL(N,C)±×R. The determinantal identity det⁡(A+B+C)=det⁡(A−1+B−1+C−1) and an extended arithmetic-Karcher-harmonic mean inequalities is derived for such triples.