# Metric regularity, strong CHIP, and CHIP are distinct properties

Bauschke, Heinz H. and Borwein, Jonathan M. and Tseng, Paul (2000) Metric regularity, strong CHIP, and CHIP are distinct properties. Journal of Convex Analysis, 7 . pp. 395-412. ISSN 0944-6532

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## Abstract

Metric 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 \emph{subspaces}. The question whether or not converse implications are true for general convex sets was open. We show that --- even for \emph{convex cones} --- the converse implications need not hold by constructing counter-examples in $\R^4$.

Item Type: Article pubdom FALSE bounded linear regularity, linear regularity, metric regularity, normal cone, property CHIP, property strong CHIP, tangent cone 90-xx Economics, operations research, programming, games > 90Cxx Mathematical programming52-xx Convex and discrete geometry > 52Axx General convexity46-xx Functional analysis > 46Cxx Inner product spaces and their generalizations, Hilbert spaces UNSPECIFIED Users 1 not found. 25 Nov 2003 21 Sep 2014 16:32 https://docserver.carma.newcastle.edu.au/id/eprint/205

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