Characterizations of subdual cones via isotonicity of the norm of the metric projection and via antitonicity of angular distance

Abstract

Let $H$ be a real Hilbert space, let $K\subseteq H$ be a closed convex cone, let $K^{\ast }$ be its dual closed convex cone, and let $P_K$ denote the metric projection onto $K$. We study scalar quantities associated with th projection $P_K$ and their interaction with the preorder induced by $K$. Our main result characterizes subduality by the monotonicity of the norm of a projection: $K\subseteq K^{\ast }$ if and only if $x{\le }_{K}y$ implies $\Vert P_{K}x\Vert \le \Vert P_{K}y\Vert$. We also discuss the sublinearity of the norm of a projection and of the distance from a closed convex cone, relate these functions to asymmetric seminorms, and introduce a normalized angular distance from a closed convex cone. The angular distance admits equivalent formulas in terms of $P_K$ and the ordinary distance function and, for closed convex cones with $K\ne H$, yields a second characterization of subduality. Finally, coordinatewise monotone norms on finite-dimensional spaces are used to aggregate distances from several closed convex cones and thereby construct asymmetric seminorms. The proofs are consequences of Moreau's decomposition theorem and the geometry of metric projections.