The Karcher mean of equilateral triangles

Abstract

In this paper we present a class of Riemannian equilateral triangles ΔABC of the Cartan-Hadamard Riemannian manifold of NXN positive definite matrices of determinant 1 whose Karcher mean of vertices is of the form ʌ(A,B,C)=A+B+C/N√det(A+B+C). The obtained class of triples (A+B+C) forms an analytic manifold with an analytic parameterization over SL(N,C)'± xR . 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.