Stability estimates of continuous selections for metric almost-projections
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.
pointwise Lipschitz in the Hausdorff metric
scientific article; zbMATH DE number 771969