The Existence of Unbounded Closed Convex Sets with Trivial Recession Cone in Normed Spaces

2015 ◽  
Vol 41 (2) ◽  
pp. 277-282
Author(s):  
Huynh The Phung
Keyword(s):  
Convex Sets ◽  
Recession Cone ◽  
Normed Spaces

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 Effective Uniform Bounds on the Krasnoselski-Mann Iteration

2000 ◽  
Vol 7 (9) ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Rate Of Convergence ◽  
Convex Sets ◽  
Fixed Point Theory ◽  
Uniformly Convex ◽  
Asymptotic Regularity ◽  
Normed Spaces ◽  
Mann Iteration ◽  
Starting Point

This paper is a case study in proof mining applied to non-effective proofs<br />in nonlinear functional analysis. More specifically, we are concerned with the<br />fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general so-called Krasnoselski-Mann iterations. These iterations converge to fixed points of f only under special compactness conditions and even for uniformly convex<br />spaces the rate of convergence is in general not computable in f (which is<br />related to the non-uniqueness of fixed points). However, the iterations yield<br />approximate fixed points of arbitrary quality for general normed spaces and<br />bounded C (asymptotic regularity).<br />In this paper we apply general proof theoretic results obtained in previous<br />papers to non-effective proofs of this regularity and extract uniform explicit<br />bounds on the rate of the asymptotic regularity. We start off with the classical<br />case of uniformly convex spaces treated already by Krasnoselski and show<br />how a logically motivated modification allows to obtain an improved bound. Already the analysis of the original proof (from 1955) yields an elementary<br />proof for a result which was obtained only in 1990 with the use of the deep<br />Browder-G¨ohde-Kirk fixed point theorem. The improved bound from the modified<br /> proof gives applied to various special spaces results which previously had<br />been obtained only by ad hoc calculations and which in some case are known<br />to be optimal.<br />The main section of the paper deals with the general case of arbitrary normed<br />spaces and yields new results including a quantitative analysis of a theorem<br />due to Borwein, Reich and Shafrir (1992) on the asymptotic behaviour of<br />the general Krasnoselski-Mann iteration in arbitrary normed spaces even for unbounded sets C. Besides providing explicit bounds we also get new qualitative results concerning the independence of the rate of convergence of the norm of that iteration from various input data. In the special case of bounded convex sets, where by well-known results of Ishikawa, Edelstein/O'Brian and Goebel/Kirk the norm of the iteration converges to zero, we obtain uniform<br />bounds which do not depend on the starting point of the iteration and the<br />nonexpansive function and the normed space X and, in fact, only depend<br />on the error epsilon, an upper bound on the diameter of C and some very general information on the sequence of scalars k used in the iteration. Even non-effectively only the existence of bounds satisfying weaker uniformity conditions was known before except for the special situation, where lambda_k := lambda is constant. For the unbounded case, no quantitative information was known so far.

scholarly journals Compactness and D-Boundedness in Menger’s 2-probabilistic normed spaces

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1263-1272 ◽  
Author(s):  
P.K. Harikrishnan ◽  
Bernardo Guillén ◽  
K.T. Ravindran
Keyword(s):  
Normed Space ◽  
Convex Sets ◽  
Normed Spaces ◽  
Probabilistic Normed Space ◽  
Probabilistic Normed Spaces

The idea of convex sets and various related results in 2-Probabilistic normed spaces were established in [7]. In this paper, we obtain the concepts of convex series closedness, convex series compactness, boundedness and their interrelationships in Menger?s 2-probabilistic normed space. Finally, the idea of D-Boundedness in Menger?s 2-probabilistic normed spaces and Menger?s Generalized 2-Probabilistic Normed spaces are discussed.

Convex sets in probabilistic normed spaces

2008 ◽  
Vol 36 (2) ◽  
pp. 322-328 ◽  
Author(s):  
Asadollah Aghajani ◽  
Kourosh Nourouzi
Keyword(s):  
Convex Sets ◽  
Normed Spaces ◽  
Probabilistic Normed Spaces

scholarly journals Average distances in compact connected spaces

1982 ◽  
Vol 26 (3) ◽  
pp. 331-342 ◽  
Author(s):  
David Yost
Keyword(s):  
Convex Sets ◽  
Average Distance ◽  
Compact Convex ◽  
Topological Spaces ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Distance Property ◽  
Compact Convex Set ◽  
Average Distance Property ◽  
Image Position

We give a simple proof of the fact that compact, connected topological spaces have the “average distance property”. For a metric space (X, d), this asserts the existence of a unique number a = a(X) such that, given finitely many points x1, …, xn ∈ X, then there is some y ∈ X withWe examine the possible values of a(X) , for subsets of finite dimensional normed spaces. For example, if diam(X) denotes the diameter of some compact, convex set in a euclidean space, then a(X) ≤ diam(X)/√2 . On the other hand, a(X)/diam(X) can be arbitrarily close to 1 , for non-convex sets in euclidean spaces of sufficiently large dimension.

scholarly journals A remark on a theorem of Browder

2013 ◽  
Vol 29 (1) ◽  
pp. 119-123
Author(s):  
CORNELIU UDREA ◽  
Keyword(s):  
Banach Space ◽  
Convex Sets ◽  
Existence Of Solutions ◽  
Type Theorem ◽  
Monotone Operators ◽  
Dual Pair ◽  
Normed Spaces ◽  
Browder’S Theorem ◽  
Weakly Continuous ◽  
Weakly Closed

This work deals with a Browder type theorem, and some of its consequences.We consider hX, Y i a dual pair of real normed spaces, C a weakly closed convex subset of X containing 0X, and L a function from C into Y which is monotone, weakly continuous on the line segments in C, and coercive. In the article ,,Nonlinear monotone operators and convex sets in Banach spaces”, Bull. Amer. Math. Soc., 71 (1965), F. E. Browder proved the existence of solutions for variational inequalities with such an operator L provided that X = E is a reflexive Banach space, and Y = E0 is its dual space. It is the object of this note to remark that a similar result is valid when Y = E is a Banach space (not necessary reflexive) and X = E0 (for example in the case of the Lebesgue spaces E = L1 (T), and E0 = L∞(T)). Moreover we shall show that the Browder’s theorem is a consequence of this result, and we shall also prove a Stampacchia type theorem.

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