Open AccessA Study of the Relationship between Convex Sets and Connected Sets
Author(s) -
Khem Raj Malla
Publication year - 2019
Publication title -
journal of advanced academic research
Language(s) - English
Resource type - Journals
eISSN - 2362-1311
pISSN - 2362-1303
DOI - 10.3126/jaar.v6i1.35311
Subject(s) - mathematics , convexity , banach space , regular polygon , social connectedness , combinatorics , space (punctuation) , differentiable function , function (biology) , counterexample , differential (mechanical device) , pure mathematics , geometry , computer science , physics , psychology , evolutionary biology , financial economics , economics , psychotherapist , biology , operating system , thermodynamics
The principal objective of this research article is to explore the relationship between convex sets and connected sets. All convex sets are connected but in all cases connected sets are not convex. In the Maly theorem,let X be a Banach space, and let f:X → R be a (Fr´echet-)differentiable function. Then, for any closed convex subset C of X with nonempty interior,the image Df(C) of C by the differential Df of f is a connected subset of X∗ , where X∗ stands for thetopological dual space of X.The result does not hold true if C has an empty interior. There are counterexamples even with functions f of two variables. This article concludes that convexity cannot be replaced with the connectedness of C.