Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization

1999 ◽  
Vol 86 (1) ◽  
pp. 135-160 ◽  
Author(s):  
Heinz H. Bauschke ◽  
Jonathan M. Borwein ◽  
Wu Li
Keyword(s):  
Convex Optimization ◽  
Error Bounds ◽  
Intersection Property ◽  
Bounded Linear ◽  
Linear Regularity ◽  
Conical Hull

Porosity and the bounded linear regularity property

2014 ◽  
Vol 20 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Simeon Reich ◽  
Alexander J. Zaslavski
Keyword(s):  
Hilbert Space ◽  
Convex Sets ◽  
Regularity Property ◽  
Nonempty Intersection ◽  
Normed Spaces ◽  
Infinite Products ◽  
Bounded Linear ◽  
Linear Regularity ◽  
Nearest Point ◽  
Convex Subsets

Abstract.H. H. Bauschke and J. M. Borwein showed that in the space of all tuples of bounded, closed, and convex subsets of a Hilbert space with a nonempty intersection, a typical tuple has the bounded linear regularity property. This property is important because it leads to the convergence of infinite products of the corresponding nearest point projections to a point in the intersection. In the present paper we show that the subset of all tuples possessing the bounded linear regularity property has a porous complement. Moreover, our result is established in all normed spaces and for tuples of closed and convex sets, which are not necessarily bounded.

scholarly journals AN EXAMPLE CONCERNING BOUNDED LINEAR REGULARITY OF SUBSPACES IN HILBERT SPACE

2013 ◽  
Vol 89 (2) ◽  
pp. 217-226 ◽  
Author(s):  
SIMEON REICH ◽  
ALEXANDER J. ZASLAVSKI
Keyword(s):  
Hilbert Space ◽  
Natural Number ◽  
Metric Spaces ◽  
Regularity Property ◽  
Finite Sets ◽  
Bounded Linear ◽  
Linear Regularity ◽  
Related Theorem

AbstractWe study bounded linear regularity of finite sets of closed subspaces in a Hilbert space. In particular, we construct for each natural number $n\geq 3$ a set of $n$ closed subspaces of ${\ell }^{2} $ which has the bounded linear regularity property, while the bounded linear regularity property does not hold for each one of its nonempty, proper nonsingleton subsets. We also establish a related theorem regarding the bounded regularity property in metric spaces.

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