Abstract

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\mathbb{R}^n into Rm\mathbb{R}^m 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.

Author information

Contact details are reproduced from the original publication and may be historical.

Jonathan M. Borwein

Faculty of Computer Science, Dalhousie University, Halifax, Nova Scotia B3H 1W5, Canada
and: School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia

jborwein@cs.dal.ca

Warren B. Moors

Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland, New Zealand

moors@math.auckland.ac.nz

Keywords

  • Closed convex cone
  • linear mapping
  • stability
  • linear programming

Mathematics Subject Classification

47N10; 90C25, 90C22

References

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Suggested citation

J. M. Borwein, W. B. Moors. “Stability of Closedness of Convex Cones under Linear Mappings.” Journal of Convex Analysis 16 (2009), No. 3&4, 699–705. https://doi.org/10.68381/jca16041

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.