Existence Of Nearest Points In Banach Spaces

1989 ◽  
Vol 41 (4) ◽  
pp. 702-720 ◽  
Author(s):  
Jonathan M. Borwein ◽  
Simon Fitzpatrick
Keyword(s):  
Banach Spaces ◽  
Closed Subset ◽  
Extension Theorem ◽  
Closed Set ◽  
Reflexive Space ◽  
Nearest Point ◽  
Open Question ◽  
Nearest Points

This paper makes a unified development of what the authors know about the existence of nearest points to closed subsets of (real) Banach spaces. Our work is made simpler by the methodical use of subderivatives. The results of Section 3 and Section 7 in particular are, to the best of our knowledge, new. In Section 5 and Section 6 we provide refined proofs of the Lau-Konjagin nearest point characterizations of reflexive Kadec spaces (Theorem 5.11, Theorem 6.6) and give a substantial extension (Theorem 5.12). The main open question is: are nearest points dense in the boundary of every closed subset of every reflexive space? Indeed can a proper closed set in a reflexive space fail to have any nearest points? In Section 7 we show that there are some non-Kadec reflexive spaces in which nearest points are dense in the boundary of every closed set.

scholarly journals NEAREST POINTS AND DELTA CONVEX FUNCTIONS IN BANACH SPACES

2015 ◽  
Vol 93 (2) ◽  
pp. 283-294
Author(s):  
JONATHAN M. BORWEIN ◽  
OHAD GILADI
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Convex Functions ◽  
Closed Set ◽  
Nearest Point ◽  
Set Of Points ◽  
Nearest Points

Given a closed set$C$in a Banach space$(X,\Vert \cdot \Vert )$, a point$x\in X$is said to have a nearest point in$C$if there exists$z\in C$such that$d_{C}(x)=\Vert x-z\Vert$, where$d_{C}$is the distance of$x$from$C$. We survey the problem of studying the size of the set of points in$X$which have nearest points in$C$. We then turn to the topic of delta convex functions and indicate how it is related to finding nearest points.

scholarly journals Nearest points to closed sets and directional derivatives of distance functions

1989 ◽  
Vol 39 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Simon Fitzpatrick
Keyword(s):  
Distance Function ◽  
Directional Derivative ◽  
Distance Functions ◽  
Directional Derivatives ◽  
Closed Set ◽  
Closed Sets ◽  
Set Of Points ◽  
Derivatives Of ◽  
Nearest Points ◽  
Dense Set

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

scholarly journals ON λ-STRICT IDEALS IN BANACH SPACES

2010 ◽  
Vol 83 (2) ◽  
pp. 231-240 ◽  
Author(s):  
TROND A. ABRAHAMSEN ◽  
OLAV NYGAARD
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Studia Math ◽  
Separable Case ◽  
Baire One ◽  
Ideal Property ◽  
Open Question

AbstractWe define and study λ-strict ideals in Banach spaces, which for λ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do λ-strict u-ideals in their biduals, at least for λ>1/2. An open question, posed by Godefroy et al. [‘Unconditional ideals in Banach spaces’, Studia Math.104 (1993), 13–59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when κu(X)=1; we prove that if κu(X)=1 then X is a strict u-ideal in Ba (X) , and we establish the converse in the separable case.

scholarly journals Tietze Extension Theorem for n-dimensional Spaces

2014 ◽  
Vol 22 (1) ◽  
pp. 11-19 ◽  
Author(s):  
Karol Pąk
Keyword(s):  
Continuous Function ◽  
Topological Space ◽  
Compact Subset ◽  
Closed Subset ◽  
Extension Theorem ◽  
Convex Compact ◽  
Arbitrary Subset ◽  
Empty Interior ◽  
Convex Compact Subset ◽  
Arbitrary Convex

Summary In this article we prove the Tietze extension theorem for an arbitrary convex compact subset of εn with a non-empty interior. This theorem states that, if T is a normal topological space, X is a closed subset of T, and A is a convex compact subset of εn with a non-empty interior, then a continuous function f : X → A can be extended to a continuous function g : T → εn. Additionally we show that a subset A is replaceable by an arbitrary subset of a topological space that is homeomorphic with a convex compact subset of En with a non-empty interior. This article is based on [20]; [23] and [22] can also serve as reference books.

On Weak Well-posedness of the Nearest Point and Mutually Nearest Point Problems in Banach Spaces

2021 ◽  
Vol 37 (8) ◽  
pp. 1303-1312
Author(s):  
Zi Hou Zhang ◽  
Chun Yan Liu ◽  
Yu Zhou ◽  
Jing Zhou
Keyword(s):  
Banach Spaces ◽  
Well Posedness ◽  
Nearest Point

scholarly journals Approximating Fixed Points of Nonself Contractive Type Mappings in Banach Spaces Endowed with a Graph

2016 ◽  
Vol 24 (2) ◽  
pp. 27-43 ◽  
Author(s):  
Laszlo Balog ◽  
Vasile Berinde ◽  
Mădălina Păcurar
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Banach Spaces ◽  
Closed Subset ◽  
Functional Equations ◽  
Fixed Point Theorems ◽  
Contraction Principle ◽  
Nonlinear Functional ◽  
Contractive Type

Abstract Let K be a non-empty closed subset of a Banach space X endowed with a graph G. We obtain fixed point theorems for nonself G-contractions of Chatterjea type. Our new results complement and extend recent related results [Berinde, V., Păcurar, M., The contraction principle for nonself mappings on Banach spaces endowed with a graph, J. Nonlinear Convex Anal. 16 (2015), no. 9, 1925-1936; Balog, L., Berinde, V., Fixed point theorems for nonself Kannan type contractions in Banach spaces endowed with a graph, Carpathian J. Math. 32 (2016), no. 3 (in press)] and thus provide more general and flexible tools for studying nonlinear functional equations.

scholarly journals Spectral Calculus and Lipschitz Extension for Barycentric Metric Spaces

2013 ◽  
Vol 1 ◽  
pp. 163-199 ◽  
Author(s):  
Manor Mendel ◽  
Assaf Naor
Keyword(s):  
Banach Spaces ◽  
Metric Spaces ◽  
Extension Theorem ◽  
Unified Framework ◽  
Lipschitz Extension

Abstract The metric Markov cotype of barycentric metric spaces is computed, yielding the first class of metric spaces that are not Banach spaces for which this bi-Lipschitz invariant is understood. It is shown that this leads to new nonlinear spectral calculus inequalities, as well as a unified framework for Lipschitz extension, including new Lipschitz extension results for CAT (0) targets. An example that elucidates the relation between metric Markov cotype and Rademacher cotype is analyzed, showing that a classical Lipschitz extension theorem of Johnson, Lindenstrauss and Benyamini is asymptotically sharp.

Intra-Class Threshold Selection in Face Space Using Set Estimation Technique

2010 ◽  
pp. 69-84
Author(s):  
Madhura Datta ◽  
C. A. Murthy
Keyword(s):  
Face Recognition ◽  
Real Life ◽  
Recognition System ◽  
Decision Threshold ◽  
Threshold Selection ◽  
Query Image ◽  
Closed Set ◽  
Set Estimation ◽  
Open Test ◽  
Open Question

Most of the conventional face recognition algorithms are dissimilarity based, and for the sake of open and closed set classification one needs to put a proper threshold on the dissimilarity value. On the basis of the decision threshold, a biometric recognition system should be in a position to accept the query image as client or reject him as imposter. However, the selection of proper threshold of a given class in a dataset is an open question, as it is related to the difficulty levels dictated in face recognition problems. In this chapter, the authors have introduced a novel thresholding technique for a real life scenario where the query face image may not be present in the training database, i.e. often referred by the biometric researchers as the open test identification. The theoretical basis of the thresholding technique and its corresponding verification on several datasets has been successfully demonstrated in the article. The proposed threshold selection is based on statistical method of set estimation and is guided by minimal spanning tree. It has been found that the proposed technique performs better than the ROC curve based threshold selection mechanism.

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