On functions of nonconvexity for graphs of continuous functions
type: Item
instance of: scholarly article
title: On functions of nonconvexity for graphs of continuous functions
zbMATH Open document ID: 0878.46014
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-01-08T00:00:00Z
review text: For a subset \(P\) of a normed linear space, the ``degree'' of non-convexity of the set \(P\) is introduced as a function \(h_P:(0,\infty)\to [0,2]\). It turns out that for a closed \(P\), the convexity of \(P\) is equivalent to \(h_P\equiv 0\). A special attention is given to the case when \(P\) is a subset of \(\mathbb{R}^2\) or a graph of a continuous function of one variable. The main application of functions \(h_P\) in the paper is that in the Michael Selection Theorem the condition of convexity can be replaced by the condition that the values of a multivalued map are graphs of polynomials \(g(x)= x^n+ a_{n-1}x^{n-1}+\cdots+ a_1x+ a_0\), \(|a_i|\leq C\).
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 861866
zbMATH Keywords: lower semicontinuity
zbMATH Keywords: degree of non-convexity
zbMATH Keywords: Michael Selection Theorem
zbMATH Keywords: convexity
zbMATH Keywords: multivalued map
zbMATH Keywords: graphs of polynomials
author: Dušan Repovš
author: Pavel V. Semenov
reviewed by: Mikhail Tkachenko
MaRDI profile type: Publication
full work available at URL: jmaa.1995.1458
OpenAlex ID: W2007081929
DOI: 10.1006/JMAA.1995.1458
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description: scientific article; zbMATH DE number 861866