  The basic tool for solving problems in metric geometry and isotonic
regression is the metric projection onto closed convex cones. Isotonicity of
these projections with respect to a given order relation can facilitate finding
the solutions of the above problems. In the recent note "A. B. N\'emeth and
S.Z. N\'emeth: Isotonic regression and isotonic projection. Linear Algebra and
its Applications, 494: 80-89 (2016)" this problem was studied for the
coordinate-wise ordering. This study was the starting point for further
investigations, such as the ones presented here. The order relation in the
Euclidean space endowed by a proper cone is considered and the proper cones
admitting isotone metric projections with respect to this order relation are
investigated.
