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.

SEPARATION OF CONVEX SETS IN EXTENDED NORMED SPACES

2015 ◽  
Vol 99 (2) ◽  
pp. 145-165 ◽  
Author(s):  
G. BEER ◽  
J. VANDERWERFF
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Separation Theorems ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Weakly Closed ◽  
Separation Of Convex Sets ◽  
Real Linear ◽  
Closed Convex Set ◽  
Special Case

We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$, that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

scholarly journals Learning Weakly Convex Sets in Metric Spaces

2021 ◽  
pp. 200-216
Author(s):  
Eike Stadtländer ◽  
Tamás Horváth ◽  
Stefan Wrobel
Keyword(s):  
Convex Sets ◽  
Metric Spaces ◽  
Weakly Convex

Properties of the Distance Function to Strongly and Weakly Convex Sets in a Nonsymmetrical Space

2020 ◽  
Vol 64 (5) ◽  
pp. 17-30
Author(s):  
S. I. Dudov ◽  
E. S. Polovinkin ◽  
V. V. Abramova
Keyword(s):  
Distance Function ◽  
Convex Sets ◽  
Weakly Convex
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