Metric properties of alternative fidelities

On the space of mixed quantum states, several alternative fidelities have been proposed besides the standard Uhlmann-Jozsa fidelity. It has been known that several properties of the Uhlmann-Jozsa fidelity still hold true for these alternative fidelities. The aim of this paper is to give positive answers to some questions about the metric properties of functionals of alternative quantum fidelities raised by Y. C. Liang {\it et al.} in \cite{LYM}. Our method is to use the non-negativity of the Gram determinant of three vectors constructed from the quantum states to prove the triangle inequality for the modified Bures angle.