Convex maps on R-n and positive definite matrices

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) bar right arrow Tr log(X* A(s) X + Y* B-t Y) is jointly convex on R-2. This is related to some exotic matrix Holder inequalities such as parallel to sinh(Sigma(m)(i=1) A(i)B(i))parallel to <= parallel to sinh(Sigma(m)(i=1) A(i)(p))parallel to(1/p)parallel to sinh(Sigma(m)(i=1) B-i(q))parallel to(1/q) for all positive matrices A(i), B-i, such that A(i)B(i) = B(i)A(i), conjugate exponents p, q and unitarily invariant norms parallel to.parallel to. 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.