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

DOI:
https://doi.org/10.1090/S0002-9947-05-03945-0

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 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:
https://doi.org/10.1090/S0002-9947-05-03945-0

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