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Journal of Convex Analysis 07 (2000), No. 2, 395--412
Copyright Heldermann Verlag 2000

Bounded Linear Regularity, Strong CHIP, and CHIP are Distinct Properties

Heinz H. Bauschke
Dept. of Mathematics and Statistics, Okanagan University College, Kelowna BC, V1V 1V7, Canada

Jonathan M. Borwein
Centre for Experimental and Constructive Mathematics, Dept. of Mathematics, Simon Fraser University, Burnaby BC, V5A 1S6, Canada

Paul Tseng
Dept. of Mathematics, University of Washington, Seattle, WA 98195-4350, U.S.A.

Bounded linear regularity, the strong conical hull intersection property (strong CHIP), and the conical hull intersection property (CHIP) are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. It was shown recently that these properties are fundamental in several branches of convex optimization, including convex feasibility problems, error bounds, Fenchel duality, and constrained approximation. It was known that regularity implies strong CHIP, which in turn implies CHIP; moreover, the three properties always hold for subspaces. The question whether or not converse implications are true for general convex sets was open.
We show that -- even for convex cones -- the converse implications need not hold by constructing counter-examples in R4.

Keywords: Bounded linear regularity, linear regularity, normal cone, property CHIP, property strong CHIP, tangent cone.

MSC: 90C25; 46C05, 52A15, 52A20

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