Stability of closedness of closed convex sets under linear mappings

Abstract

We study the problem of when the continuous linear image of a fixed closed convex set $X\subset {\mathbb{R}}^n$ is closed. Specifically, we improve the main results in the papers \cite{Borwein2009, Borwein2010} by showing that for all, except for at most a $\sigma$-porous set, of the linear mappings $T$ from ${\mathbb{R}}^n$ into ${\mathbb{R}}^m,$ not only $T(X)$ is closed, but there is also an open neighborhood of $T$ whose members also preserve the closedness of $X.$