Paraconvexity of the graphs of Lipschitzian functions
type: Item
instance of: scholarly article
title: Paraconvexity of the graphs of Lipschitzian functions
zbMATH Open document ID: 0882.54019
DOI: 10.1007/BF02362519
published in: Journal of Mathematical Sciences (New York)
publication date: 1996-12-10T00:00:00Z
review text: Following \textit{E. Michael} [Math. Scand. 7, 372-376 (1960; Zbl 0093.12001)] a closed subset \(P\) of a Banach space \(B\) is called \(\alpha\)-paraconvex if for \(x\in B\), \(r> \text{dist} (x,P)\) and \(y\in\text{conv} (P\cap K(x,r))\) we have \(\text{dist} (y,P)\leq \alpha \cdot r\), where \(K(x,r): =\{z\in B:|z-x |\leq r\}\). Each lsc multifunction defined on a paracompact space \(T\) and having \(\alpha\)-paraconvex values \(F(t) \subset B\) admits a continuous selection \(f:T\to B\) if \(0\leq \alpha<1\). In the paper under review it is proved, by using direct geometrical considerations, that if \(B=X \oplus Y\), where \(x=\mathbb{R}^n\), \(Y=\mathbb{R}\), then the graphs of \(k\)-Lipschitzian functions \(g:X\to Y\) with the same constant \(k\) are \(\alpha \)-paraconvex and an explicit formula for \(\alpha\) is given. The paper contains four questions concerning the optimisation of the values of \(\alpha\) and the possibility of replacing \(X\) and \(Y\) by arbitrary Hilbert spaces, or \(Y\) by \(\mathbb{R}^n\). Reviewers notes: 1. A. Maliszewski in a private communication to the reviewer indicates the following counterexample: Let \(f:\mathbb{R} \to \mathbb{R}^2\), with the sup norm, be such that \(f(-3)= (-3,3)\), \(f(-1)= (-3,-3)\), \(f(1)= (3,-3)\), \(f(3)= (3,3)\), \(f\) periodic with the period 8 and linear in the segments \([-3,-1]\), \([-1,1]\), \([1,3]\) and \([3,5]\). Then \(f\) is Lipschitzian with \(k=3\), but the graph of \(f\) is not \(\alpha\)-paraconvex. Nevertheless a multifunction \(F:\mathbb{R}^n \to\mathbb{R}^3\), whose values are graphs of \(f\in\text{Lip}_3 (\mathbb{R}, \mathbb{R}^2)\) has continuous selection. 2. The paper contains some misprints: \(e_1, \dots, e_n\) cannot create a base for \((n+1)\)-dimensional \(V_{n+1}\) on p. 239 and for the set \(W\) on the same page the assumption of closedness is necessary.
reviewed by: Włodzimierz Ślęzak
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 26A16
Mathematics Subject Classification ID: 52A30
zbMATH DE Number: 955861
zbMATH Keywords: paraconvex set
zbMATH Keywords: multifunction
zbMATH Keywords: Lipschitz map
author: Pavel V. Semenov
MaRDI profile type: Publication
cites work: Continuous selections. I
cites work: Paraconvex Sets.
cites work: Cell-like maps. The problem of raising dimension
full work available at URL: https://doi.org/10.1007/bf02362519
OpenAlex ID: W2036615975
description: scientific article; zbMATH DE number 955861