In this paper, we give some new conditions for the existence and uniqueness of mutually nearest points of two sets, i.e., two points which achieve the minimum distance between two sets in a metric or Banach spaces. These conditions are seen in case of compact sets, weak Compact sets, closed and convex sets, weakly sequentially compact sets and boundedly compact sets and their combinations. The study is confined to metric spaces and normed or Banach spaces. Some geometric properties of a Banach spaces like; strictly convexity, uniformly convexity, P-Property and weak P-Property are introduced. Also, we introduce the concept of generalized weak P-Property and give some interesting results. The present work may be briefly outlined as follows: It is the mathematical study that is motivated by the desire to seek answers to the following basic questions, among others. Which subsets are mutually proximinal? How does one recognize when given elements x ∈ A and y ∈ B are the nearest points of A and B? which is called a natural extension of the best approximation problem. Can one describe some useful algorithms for actually computing nearest points between two given sets? And how to find closely related sets to the proximity maps.
| Published in | International Journal of Theoretical and Applied Mathematics (Volume 4, Issue 3) |
| DOI | 10.11648/j.ijtam.20180403.11 |
| Page(s) | 29-34 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2018. Published by Science Publishing Group |
Nearest Point, P-Property, Weak P-Property, Compact Set, Minimizing Sequence, Generalized Weak P-Property
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APA Style
Gaj Ram Damai, Prakash Muni Bajracharya. (2018). Mutually Nearest Points for Two Sets in Metric Spaces. International Journal of Theoretical and Applied Mathematics, 4(3), 29-34. https://doi.org/10.11648/j.ijtam.20180403.11
ACS Style
Gaj Ram Damai; Prakash Muni Bajracharya. Mutually Nearest Points for Two Sets in Metric Spaces. Int. J. Theor. Appl. Math. 2018, 4(3), 29-34. doi: 10.11648/j.ijtam.20180403.11
AMA Style
Gaj Ram Damai, Prakash Muni Bajracharya. Mutually Nearest Points for Two Sets in Metric Spaces. Int J Theor Appl Math. 2018;4(3):29-34. doi: 10.11648/j.ijtam.20180403.11
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Download@article{10.11648/j.ijtam.20180403.11,
author = {Gaj Ram Damai and Prakash Muni Bajracharya},
title = {Mutually Nearest Points for Two Sets in Metric Spaces},
journal = {International Journal of Theoretical and Applied Mathematics},
volume = {4},
number = {3},
pages = {29-34},
doi = {10.11648/j.ijtam.20180403.11},
url = {https://doi.org/10.11648/j.ijtam.20180403.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20180403.11},
abstract = {In this paper, we give some new conditions for the existence and uniqueness of mutually nearest points of two sets, i.e., two points which achieve the minimum distance between two sets in a metric or Banach spaces. These conditions are seen in case of compact sets, weak Compact sets, closed and convex sets, weakly sequentially compact sets and boundedly compact sets and their combinations. The study is confined to metric spaces and normed or Banach spaces. Some geometric properties of a Banach spaces like; strictly convexity, uniformly convexity, P-Property and weak P-Property are introduced. Also, we introduce the concept of generalized weak P-Property and give some interesting results. The present work may be briefly outlined as follows: It is the mathematical study that is motivated by the desire to seek answers to the following basic questions, among others. Which subsets are mutually proximinal? How does one recognize when given elements x ∈ A and y ∈ B are the nearest points of A and B? which is called a natural extension of the best approximation problem. Can one describe some useful algorithms for actually computing nearest points between two given sets? And how to find closely related sets to the proximity maps.},
year = {2018}
}
TY - JOUR T1 - Mutually Nearest Points for Two Sets in Metric Spaces AU - Gaj Ram Damai AU - Prakash Muni Bajracharya Y1 - 2018/11/07 PY - 2018 N1 - https://doi.org/10.11648/j.ijtam.20180403.11 DO - 10.11648/j.ijtam.20180403.11 T2 - International Journal of Theoretical and Applied Mathematics JF - International Journal of Theoretical and Applied Mathematics JO - International Journal of Theoretical and Applied Mathematics SP - 29 EP - 34 PB - Science Publishing Group SN - 2575-5080 UR - https://doi.org/10.11648/j.ijtam.20180403.11 AB - In this paper, we give some new conditions for the existence and uniqueness of mutually nearest points of two sets, i.e., two points which achieve the minimum distance between two sets in a metric or Banach spaces. These conditions are seen in case of compact sets, weak Compact sets, closed and convex sets, weakly sequentially compact sets and boundedly compact sets and their combinations. The study is confined to metric spaces and normed or Banach spaces. Some geometric properties of a Banach spaces like; strictly convexity, uniformly convexity, P-Property and weak P-Property are introduced. Also, we introduce the concept of generalized weak P-Property and give some interesting results. The present work may be briefly outlined as follows: It is the mathematical study that is motivated by the desire to seek answers to the following basic questions, among others. Which subsets are mutually proximinal? How does one recognize when given elements x ∈ A and y ∈ B are the nearest points of A and B? which is called a natural extension of the best approximation problem. Can one describe some useful algorithms for actually computing nearest points between two given sets? And how to find closely related sets to the proximity maps. VL - 4 IS - 3 ER -
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