Stability estimates of continuous selections for metric almost-projections
type: Item
instance of: scholarly article
title: Stability estimates of continuous selections for metric almost-projections
zbMATH Open document ID: 0880.46014
DOI: 10.1007/BF02112475
author: A. V. Marinov
published in: Mathematical Notes
publication date: 1995-07-10T00:00:00Z
review text: Let \((X,|\cdot|)\) be a normed linear space (NLS), \(\varepsilon\geq 0\). A metric almost-projection or \(\varepsilon\)-projection of an element \(x\in X\) on a subset \(M\subset X\) is the set \[ x^\varepsilon_M= \Biggl\{z\in M: |z-x|\leq\inf_{y\in M} |y- x|+\varepsilon\Biggr\}. \] For \(\varepsilon= 0\) we denote this projection by \(x_M\). Let \(Y\) be a subspace of \(X\). We denote by \(C_V(Y)\) the family of all nonempty convex closed subsets of \(Y\) and by \(\mathbb{R}_+(\mathbb{R}_0)\) the set of positive (nonnegative) real numbers. We put \(Z_0(Y)= X\times C_V(Y)\times\mathbb{R}_0\), \(Z(Y)= X\times C_V(Y)\times\mathbb{R}_+\). A mapping \(P:Z_0(X)\to C_V(X)\) given by the equality \(P(x,M,\varepsilon)= x^\varepsilon_M\) is called an operator of metric almost-projection. In the present paper, we prove the existence and obtain stability estimates for the continuous selection of the operator \(P\) which acts on the space \((Z(Y),\rho)\). We use the stability estimate for the operator \((x,M)\to x_M\) with the assumption that the space \(X\) is uniformly convex (Theorem 2). What is more, the key role in the proofs is played by the inequality obtained in [the author, Collect. Artic. Sverdlovsk 1985, 105-117 (1985; Zbl 0619.65049)], which, in particular, implies that the operator \(P\) is pointwise Lipschitz in the Hausdorff metric.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 41A65
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H04
zbMATH DE Number: 771969
zbMATH Keywords: selection
zbMATH Keywords: set-valued mapping
zbMATH Keywords: metric almost-projection
zbMATH Keywords: \(\varepsilon\)-projection
zbMATH Keywords: operator of metric almost-projection
zbMATH Keywords: stability estimate
zbMATH Keywords: pointwise Lipschitz in the Hausdorff metric
MaRDI profile type: Publication
cites work: A survey of metric selections
cites work: Characterizations of continuous and Lipschitz continuous metric selections in normed linear spaces
cites work: Set valued mappings, continuous selections, and metric projections
cites work: Continuous selections. I
cites work: Q3225964
cites work: Q3887261
cites work: The continuity of metric projections as functions of the data
cites work: The Steiner Point of a Convex Polytope
full work available at URL: https://doi.org/10.1007/bf02112475
OpenAlex ID: W2085279157
description: scientific article; zbMATH DE number 771969