Strong CHIP, normality, and linear regularity of convex sets

Bakan, Andrew; Deutsch, Frank; Li, Wu

doi:10.1090/S0002-9947-05-03945-0

Strong CHIP, normality, and linear regularity of convex sets
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by Andrew Bakan, Frank Deutsch and Wu Li

Trans. Amer. Math. Soc. 357 (2005), 3831-3863

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

Published electronically: May 10, 2005

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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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