Journal of Convex Analysis 28 (2021), No. 4, 1281--1291
Copyright Heldermann Verlag 2021
Stability of Closedness of Closed Convex Sets under Linear Mappings
Si Tiep Dinh
Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
Department of Mathematics, Dalat University, Dalat, Vietnam
We study the problem of when the linear image of a fixed closed convex subset X of Rn 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 Rn into Rm, not only T(X) is closed, but there is also an open neighborhood of T whose members also preserve the closedness of X.
Keywords: Asymptotic cone, closedness, convex cone, convex set, linear mapping, stability, sigma-porosity.
MSC: 47N10; 90C25, 90C22.
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