| Summary: | We obtain several convexity statements involving positive definite matrices. In particular, if $A,B,X,Y$ are invertible matrices and $A,B$ are positive, we show that the map
\[ (s,t) \mapsto \mathrm{Tr}\,\log \left(X^*A^sX + Y^*B^tY\right) \]
is jointly convex on $\mathbb{R}^2$. This is related to some exotic matrix Hölder inequalities such as
\[ \left\Vert \sinh \left(\sum _{i=1}^m A_iB_i\right) \right\Vert \le \left\Vert \sinh \left(\sum _{i=1}^m A_i^p\right) \right\Vert ^{1/p} \left\Vert \sinh \left(\sum _{i=1}^m B_i^q\right) \right\Vert ^{1/q} \]
for all positive matrices $A_i, B_i$, such that $A_iB_i=B_iA_i$, conjugate exponents $p,q$ and unitarily invariant norms $\Vert \cdot \Vert $. Our approach to obtain these results consists in studying the behaviour of some functionals along the geodesics of the Riemanian manifold of positive definite matrices.
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