Subadditive retractions on cones and asymmetric vector norms

Mutually polar retractions on convex cones

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.

内积格正锥的最佳逼近

Best approximation from the positive cone of an inner product lattice

本文证明了内积有序向量空间中向量格结构与正锥度量投影的保序次可加性之间的等价关系，并将最佳逼近与正部的一致性推广到非完备情形。

Geometry of the proper asymmetric norm

There is given the geometric characterization of an asymmetric norm $q$ on the real vector space $X$, for which exists an $u\in X$ such that $q(x-q(x)u)=0$, for each $x\in X$. The result is used in the theory of mutually polar retractions on cones.