Stability of Closedness of Closed Convex Sets under Linear Mappings

We study the problem of when the linear image of a fixed closed convex subset X of $\mathbb{R}^n$ is closed. Specifically, we improve results of J. M. Borwein and W. B. Moors [Stability of closedness of convex cones under linear mappings I, J. Convex Analysis 16 (2009) 699–705; Stability of closedness of convex cones under linear mappings II, J. Nonlinear Analysis Optim. 1 (2010) 1–7] by showing that for almost all 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.