Geometric properties of positive definite matrices cone with respect to the Thompson metric (Q551328)

The \textit{generalized geometric mean} of two positive definite matrices \(A\) and \(B\) is defined to be \(A \sharp_s B:= A^{\frac 12} \big( A^{-\frac 12}BA^{-\frac 12} \big)^s A^{\frac 12}\). \(A\sharp B:= A\sharp_{\frac 12} B\) is the well-known geometric mean. The authors discuss geometric properties of a quadrangle with parallelogramic properties in a convex cone of positive definite matrices with respect to the Thompson metric as introduced by \textit{A. C. Thompson} [Proc. Am. Math. Soc. 14, 438--443 (1963; Zbl 0147.34903)].