Moreau-Type Characterizations of Polar Cones

A theorem of Moreau (1962) states that given a closed convex cone $C$ and its (negative) polar cone $C^{\circ }$ in a real Hilbert space $H$, vectors $y\in C$ and $z\in C^{\circ }$ are metric projections of a vector $u\in H$ on $C$ and $C^{\circ }$, respectively, if and only if they satisfy the following conditions: $y$ and $z$ are orthogonal and $u=y+z$. We show that these conditions provide characteristic properties of polar cones $C$ and $C^{\circ }$ in the family of pairs of convex subsets of $H$ or ${\mathbb{R}}^n$. A related result on separation of $C$ a face of $C^{\circ }$ in ${\mathbb{R}}^n$ is proved.