Trace inequalities and kinematic metrics

Kinematics remains one of the cornerstones of robotics, and over the decade, Robotica has been one of the venues in which groundbreaking work in kinematics has always been welcome. A number of works in the kinematics community have addressed metrics for rigid-body motions in multiple different venues. An essential feature of any distance metric is the triangle inequality. Here, relationships between the triangle inequality for kinematic metrics and so-called trace inequalities are established. In particular, we show that the Golden-Thompson inequality (a particular trace inequality from the field of statistical mechanics) which holds for Hermitian matrices remarkably also holds for restricted classes of real skew-symmetric matrices. We then show that this is related to the triangle inequality for $SO(3)$ and $SO(4)$ metrics.