This question is related to this recent one.
Recall that on the cone $SPD_n$ of $n\times n$ symmetric positive definite matrices, there are various notions of means, which extend the same notions already known for positive scalars: arithmetic, geometric, harmonic. In addition, they satisfy the same order: $$m_H(A,B)\le m_G(A,B)\le m_A(A,B).$$
I wander whether the following kind of mean is known. I call it the cofactor-mean, denoted $m_C(A,B)$, but shall be happy to adopt a well-established terminology.
Let $A\mapsto \hat A$ denote the cofactor map, and $B\mapsto \check B$ its inverse. For positive definite symmetric matrices, $\hat A=(\det A)A^{-1}$ and $\check B=(\det B)^{\frac1{n-1}}B^{-1}$. The cofactor-mean is $$m_C(A,B)=\widehat{\frac12(\check A+\check B)}.$$ (Apology: the widehat is not wide enough, it should cover the fraction $\frac12\,$.)
It satisfies the following inequality $$m_H(A,B)\le m_C(A,B).$$
