On stability of the metric projection operator (Q2840364)

The authors study stability properties of metric projections \( P_M \) in terms of NEWLINE\[NEWLINEd(M,N)=\max\left\{{\sup_{m\in M}{\| m\| =1}}\inf_{n\in N}\| m-n\|,\quad {\sup_{n\in N}{\| n\| =1}}\inf_{m\in M}\| m-n\|\right\}.NEWLINE\]NEWLINE Firstly, they consider a closed linear subspace \( M \) of the normed linear space \( X \) such that the metric projection \( P_M \), \( \| P_M f-f\|=\inf_{m\in M}\| m-f\| \), satisfies the strong uniqueness condition of order \( \alpha>0 \), i.e., for each \( f\in X \) and every \( m\in M \) NEWLINENEWLINE\[NEWLINE\gamma_M(f)\| P_Mf-m\|^{\alpha}\leq\| f-m\|^{\alpha}-\| f-P_Mf\|^{\alpha}NEWLINE\]NEWLINE with some constant \( \gamma_M(f)>0 \). Then, for any \( f\in X \) and every other closed linear subspace \( N\subset X \) holds the inequalityNEWLINENEWLINE\[NEWLINE\| P_Mf-P_Nf\|\leq 10\gamma_M(f)^{-1/\alpha}\| f\| d(M,N)^{1/\alpha}NEWLINE\]NEWLINE NEWLINE(Theorem 2.1). Better stability estimates can be proved in \( L^p \). Here, the main result (Theorem 3.4) is as follows:NEWLINENEWLINENEWLINEConsider the space \( L^p(K,\mu)\), \(p>2\), where \( \mu \) is a nonatomic measure. Let \( M \) be an \(r\)-dimensional subspace of \( L^p(K,\mu) \) satisfying the \( Z_{\mu} \) property, i.e., \( \mu\{x:m(x)=0\}=0 \) for every \( m\in M\backslash\{0\} \). Then, for every \( f\in L^p(K,\mu)\), \( p>2\), there exists a constant \( c_{M,f} \) such that for any \(r\)-dimensional subspace \( N \) of \( L^p(K,\mu)\), \(p>2\), the inequality NEWLINE\[NEWLINE\| P_Mf-P_Nf\|_{p}\leq c_{M,f}d(M,N)NEWLINE\]NEWLINE is valid. Furthermore, the \( Z_{\mu} \) property of \( M \) is necessary for the above estimate to hold.NEWLINENEWLINEThe case of subspaces of co-dimension 1 is studied in details.