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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Strong CHIP, normality, and linear regularity of convex sets


Authors: Andrew Bakan, Frank Deutsch and Wu Li
Journal: Trans. Amer. Math. Soc. 357 (2005), 3831-3863
MSC (2000): Primary 90C25, 41A65; Secondary 52A15, 52A20, 41A29
Published electronically: May 10, 2005
MathSciNet review: 2159690
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Abstract: We extend the property (N) introduced by Jameson for closed convex cones to the normal property for a finite collection of convex sets in a Hilbert space. Variations of the normal property, such as the weak normal property and the uniform normal property, are also introduced. A dual form of the normal property is derived. When applied to closed convex cones, the dual normal property is the property (G) introduced by Jameson. Normality of convex sets provides a new perspective on the relationship between the strong conical hull intersection property (strong CHIP) and various regularity properties. In particular, we prove that the weak normal property is a dual characterization of the strong CHIP, and the uniform normal property is a characterization of the linear regularity. Moreover, the linear regularity is equivalent to the fact that the normality constant for feasible direction cones of the convex sets at $x$ is bounded away from 0 uniformly over all points in the intersection of these convex sets.


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Additional Information

Andrew Bakan
Affiliation: Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv 01601, Ukraine
Email: andrew@bakan.kiev.ua

Frank Deutsch
Affiliation: Department of Mathematics, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: deutsch@math.psu.edu

Wu Li
Affiliation: NASA Langley Research Center, Hampton, Virginia 23681
Email: w.li@nasa.gov

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03945-0
PII: S 0002-9947(05)03945-0
Keywords: Moreau-Rockafellar equality, Jameson's property (N), Jameson's property (G), the conical hull intersection property (the CHIP), the strong conical hull intersection property (the strong CHIP), basic constraint qualification, linear regularity, bounded linear regularity, normal property, weak normal property, uniform normal property, dual normal property.
Received by editor(s): May 30, 2002
Published electronically: May 10, 2005
Article copyright: © Copyright 2005 American Mathematical Society