(eng): This paper sheds some new light on projection quantiles. It studies the subgradient conditions associated with these quantiles in a general regression context, introduces Lagrange multipliers with rich interpretation, provides another proof that corresponding projection quantile regions coincide with the halfspace depth ones, and shows how this equivalence could be used for exact computation of the latter regions by means of projection quantiles. Furthermore, it demonstrates that the regression quantile regions introduced in Hallin, Paindaveine and Siman (2010) can also be obtained from projection (regression) quantiles, which may lead to a faster computation of those regions in some particular cases.