Journal of Convex Analysis 16 (2009), No. 3, 699--705
Copyright Heldermann Verlag 2009
Stability of Closedness of Convex Cones under Linear Mappings
Jonathan M. Borwein
Faculty of Computer Science, Dalhousie University, Halifax, Nova Scotia B3H 1W5, Canada
Warren B. Moors
Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand
We reconsider the question of when the continuous linear image of a closed convex cone is closed in Euclidean space. In particular, we show that although it is not true that the closedness of the image is preserved under small perturbations of the linear mappings it is "almost" true that the closedness of the image is preserved under small perturbations, in the sense that, for "almost all" linear mappings from Rn into Rm if the image of the cone is closed then there is a small neighbourhood around it whose members also preserve the closedness of the cone.
Keywords: Closed convex cone, linear mapping, stability, linear programming.
MSC: 47N10; 90C25, 90C22
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