On sums and convex combinations of projectors onto convex sets

TL;DR

This paper characterizes when the projector onto the Minkowski sum of convex sets equals the sum of individual projectors, extending known results and analyzing convex combinations and cones.

Contribution

It provides a complete characterization of when projectors onto Minkowski sums of convex sets equal sums of individual projectors, extending previous results.

Findings

Conditions for equality of projectors onto Minkowski sums

Partial sum property for convex cones

Detailed univariate case analysis

Abstract

The projector onto the Minkowski sum of closed convex sets is generally not equal to the sum of individual projectors. In this work, we provide a complete answer to the question of characterizing the instances where such an equality holds. Our results unify and extend the case of linear subspaces and Zarantonello's results for projectors onto cones. A detailed analysis in the case of convex combinations is also carried out. We establish the partial sum property for projectors onto convex cones, and we also present various examples as well as a detailed analysis in the univariate case.