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

If K and L are mutually dual pointed convex cones in R n with the metric projections onto them denoted by P K and P L respectively, then the following two assertions are equivalent: (i) P K is isotone with respect to the order induced by K (i.e. v − u ∈ K implies P K v − P K u ∈ K ); (ii) P L is subadditive with respect to the order induced by L (i.e. P L u + P L v − P L ( u + v ) ∈ L for any u , v ∈ R n ).