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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

Tien-Son Pham
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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