Mutually polar retractions on convex cones

TL;DR

This paper explores the conditions under which mutually polar retractions exist on convex cones in Banach spaces, focusing on their properties and the case when one cone is one-dimensional.

Contribution

It characterizes the existence of mutually polar retractions on convex cones and analyzes their subadditivity in specific dimensional cases.

Findings

Mutually polar retractions exist under certain conditions.

When one cone is one-dimensional, retractions are subadditive.

The paper provides theoretical insights into the structure of retractions on convex cones.

Abstract

Two retractions Q and R on closed convex cones M and respectively N of a Banach space are called mutually polar if Q+R=I and QR=RQ=0. This note investigates the existence of a pair of mutually polar retractions for given cones M and N. It is shown that if dim N=1 (or dim M=1) then the retractions are subadditive with respect to the order relation their cone range endow.