Moreau-Type Characterizations of Polar Cones

TL;DR

This paper extends Moreau's theorem by characterizing polar cones through orthogonality and projection properties, offering new insights into convex analysis and separation in Hilbert spaces and Euclidean spaces.

Contribution

It generalizes Moreau's characterization of polar cones to broader convex set pairs and explores separation properties of cones and their faces.

Findings

Characterization of polar cones via orthogonality and projections.

Extension of Moreau's theorem to convex set pairs.

Results on separation of cones and faces in Euclidean space.

Abstract

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.