In the paper [A Riemannian framework for tensor computing](https://doi.org/10.1007/s11263-005-3222-z "Pennec, X., Fillard, P. & Ayache, N. Int J Comput Vision 66, 41–66 (2006). zbMATH review at https://zbmath.org/?q=an:1287.53031"), by Pennec et al., on page 46 the authors state a "distance" function on the manifold of positive definite matrices $\mathcal{Sym}\_n^+$ given by


$$d(A,B) = \lVert\log A^{-1/2}BA^{-1/2}\rVert$$
where the norm on the right is the standard sum of squares euclidean norm (presumably on the eigenvalues of the argument to the log-term).


The authors then say that they could not prove (but had strong empirical evidence) that the above function $d$ satisfies the triangle inequality. I am not that familiar with differential geometry, so I don't know where to check whether this property holds (I need it in an application).


Where should I look for a proof / disproof of the triangle inequality for $d$?



Proving triangle inequality for an affine invariant distance on $\mathcal{Sym}_n^+$
Let $X$, $Y$ and $Z$ be positive definite Hermitian matrices (you ask about the real symmetric case, the Hermitian one includes it). Let the eigenvalues of $(X^\*)^{-1/2} Y X^{-1/2}$ be $e^{\alpha\_i}$, those of $(Y^\*)^{-1/2} Z Y^{-1/2}$ be $e^{\beta\_i}$ and those of $(X^\*)^{-1/2} Z X^{-1/2}$ be $e^{\gamma\_i}$. Set $A=Y^{1/2} X^{-1/2}$, $B=Z^{1/2} Y^{-1/2}$ and $C=Z^{1/2} X^{-1/2}$. The [singular values](http://en.wikipedia.org/wiki/Singular_value) of $A$ are then $e^{\alpha\_i/2}$, and so forth. 


Note that $AB=C$. By a [result of Klyachko](http://www.math.neu.edu/~suciu/GAS/klyachko/kl.pdf), there exist Hermitian matrices $\mathfrak{a}$, $\mathfrak{b}$ and $\mathfrak{c}$ such that $\mathfrak{a}+\mathfrak{b}=\mathfrak{c}$, the eigenvalues of $\mathfrak{a}$ are $\alpha\_i/2$, those of $\mathfrak{b}$ are $\beta\_i/2$ and those of $\mathfrak{c}$ are $\gamma\_i/2$. This result can be thought of as saying that, although $\log A + \log B \neq \log C$, and although $\log A$, $\log B$ and $\log C$ are not Hermitian, we can find matrices which do have those properties and have the same eigenvalues. 


The inequality you want to prove is that 
$$\left( \sum \alpha\_i^2 \right)^{1/2} + \left( \sum \beta\_i^2 \right)^{1/2} \geq \left( \sum \gamma\_i^2 \right)^{1/2}$$


But $\sum \alpha\_i^2 = 4 \mathrm{Tr} \ \mathfrak{a}^\* \mathfrak{a}$. So this turns into the (standard) fact that $\mathrm{Tr} \ \mathfrak{a}^\* \mathfrak{a}$ is a positive definite norm on the Hermitian matrices. 




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I can't resist indulging in a little self promotion. This argument appears at the beginning of my paper [Horn's Problem, Vinnikov Curves and the Hive Cone](http://arxiv.org/abs/math.AG/0311428). There I consider the curve $\det(xX+yY+zZ)=0$ in $\mathbb{RP}^2$. This curve is hyperbolic and, by results of Helton and Vinnikov, all hyperbolic curves are of this form. It meets the three coordinate lines precisely at $-e^{\alpha\_i}$, $-e^{\beta\_i}$ and $-e^{\gamma\_i}$. The point is to use results from the theory of hyperbolic curves to explain Horn's results on eigenvalues of matrix sums.


