Abstract and Applied Analysis
Volume 2005 (2005), Issue 3, Pages 319-326
doi:10.1155/AAA.2005.319
A porosity result in convex minimization
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.