A duality between the metric projection onto a convex cone and the metric projection onto its dual

If KK and LL are mutually dual pointed convex cones in RnRn with the metric projections onto them denoted by PKPK and PLPL respectively, then the following two assertions are equivalent: (i) PKPK is isotone with respect to the order induced by KK (i.e. v−u∈Kv−u∈K implies PKv−PKu∈KPKv−PKu∈K); (ii) PLPL is subadditive with respect to the order induced by LL (i.e. PLu+PLv−PL(u+v)∈LPLu+PLv−PL(u+v)∈L for any u,v∈Rn).