Volume 4, Issue 3, June 2018, Page: 29-34
Mutually Nearest Points for Two Sets in Metric Spaces
Gaj Ram Damai, Department of Mathematics, Tribhuvan University, Siddhnath Science Campus, Mahendranagar, Nepal
Prakash Muni Bajracharya, Central Department of Mathematics, Tribhuvan University, Kirtipur, Kathmandu, Nepal
Received: Aug. 27, 2018;       Accepted: Sep. 18, 2018;       Published: Nov. 7, 2018
DOI: 10.11648/j.ijtam.20180403.11      View  156      Downloads  20
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.
Keywords
Nearest Point, P-Property, Weak P-Property, Compact Set, Minimizing Sequence, Generalized Weak P-Property
To cite this article
Gaj Ram Damai, Prakash Muni Bajracharya, Mutually Nearest Points for Two Sets in Metric Spaces, International Journal of Theoretical and Applied Mathematics. Vol. 4, No. 3, 2018, pp. 29-34. doi: 10.11648/j.ijtam.20180403.11
Copyright
Copyright © 2018 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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