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

Abstract

If $K$ and $L$ are mutually dual closed convex cones in a Hilbert space 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. $v-u\in K$ implies $P_Kv-P_Ku\in K$); (ii) $P_L$ is subadditive with respect to the order induced by $L$ (i. e. $P_Lu+P_Lv-P_L(u+v)\in L$ for any $u, v \in \R^n$). This extends the similar result of A. B. N\'emeth and the author for Euclidean spaces. The extension is essential because the proof of the result for Euclidean spaces is essentially finite dimensional and seemingly cannot be extended for Hilbert spaces. The proof of the result for Hilbert spaces is based on a completely different idea which uses extended lattice operations.