Relaxed triangle inequality for the orbital similarity criterion by Southworth and Hawkins and its variants

In this article, we prove the relaxed triangle inequality for Southworth and Hawkins, Drummond and Jopek orbital similarity criteria on the set of non-rectilinear Keplerian orbits with the eccentricity bounded above. We give estimates of the minimal coefficients in the inequality for each criterion and show that one of the calculated coefficients is exactly minimal. The obtained inequalities can be used for the acceleration of algorithms involving pairwise distances calculations between orbits. We present an algorithm for calculation of all distances not exceeding a fixed number in a quasi-metric space and demonstrate that the algorithm is faster than the complete calculation on the set of meteors orbits. Finally, we estimate the correlation dimensions of the set of main belt asteroids orbits and meteors orbits with respect to various orbital metrics and quasi-metrics.