Abstract and Applied Analysis
Volume 2005 (2005), Issue 3, Pages 319-326
doi:10.1155/AAA.2005.319

A porosity result in convex minimization

P. G. Howlett1 and A. J. Zaslavski2

1Centre for Industrial and Applied Mathematics (CIAM), University of South Australia, Mawson Lakes, 5059, SA, Australia
2Department of Mathematics, Mathematics, Technion – Israel Technology Institute, Haifa 32000, Israel

Received 1 August 2003

Copyright © 2005 P. G. Howlett and A. J. Zaslavski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the minimization problem f(x)→min, x∈C, where f belongs to a complete metric space ℳ of convex functions and the set C is a countable intersection of a decreasing sequence of closed convex sets Ci in a reflexive Banach space. Let ℱ be the set of all f∈ℳ for which the solutions of the minimization problem over the set Ci converge strongly as i→∞ to the solution over the set C. In our recent work we show that the set ℱ contains an everywhere dense Gδ subset of ℳ. In this paper, we show that the complement ℳ\ℱ is not only of the first Baire category but also a σ-porous set.