Regular retractions onto finite dimensional convex sets and the AR-property for Roberts spaces

Abstract

It is proved that if $X$ is an $n$-dimensional closed convex subset in a linear metric space $E$, then there is a retraction $r:E\rightarrow X$ such that $\Vert x-r(x)\Vert\leq 2(n+1)\Vert x-X\Vert$ for every $x\in E$. This fact is applied to study the AR-property in linear metric spaces. We identify a class of Roberts spaces with the AR-property. We also give a direct proof that for every $p\in[0,1 )$,$L_{\rho}$ is a needle point space.