  We present several theorems on strict and strong convexity, and higher order
differential formulae for sandwiched quasi-relative entropy (a parametrised
version of the classical fidelity). These are crucial for establishing global
linear convergence of the gradient projection algorithm for optimisation
problems for these functions. The case of the classical fidelity is of special
interest for the multimarginal optimal transport problem (the $n$-coupling
problem) for Gaussian measures.
