Optimal sunny selections for metric projections onto unit balls
Let \(X\) be a real normed vector space of dimension greater than 1, \(Ca\) empty closed convex subset of \(X\) and \({\mathcal P}: x\to 2^C\) the metric projection onto \(C\) is \({\mathcal P} (x)= \{z\in C: |x-z |= \inf_{y\in C} |x-y |\}\). The optimal Lipschitz constant of \({\mathcal P}\) is defined by \(K_{rp} (x)= \inf K_p(x)\), where the infimum is taken over all selections \(P\) of \({\mathcal P}\) and \(K_p(x)\) is the best Lipschitz constant of \(P\) defined by \(K_p (x)= \sup\{{|Px-Py |\over |x-y |}: x\neq y\}\). A metric projection \(T\) of \({\mathcal P}\) is said to be optimal if \(K_T(x) =K_{\mathcal P} (x)\). If \(C(Q)\) is the Banach space of all continuous real-valued functions on a compact Hausdorff space \(Q\) equipped with the uniform norm \(|x|_\infty= \sup_{s\in Q} |x(s)|\) and \(B_\infty= \{x\in C(Q): |x |_\infty \leq 1\}\) is its unit ball then a selection \(P\) of the metric projection \({\mathcal P}: C(Q)\to 2^{B_\infty}\) is called sunny if \(P_{x_\alpha} =P_x\) for all \(x\in C(Q)\) and \(d\geq 0\), where \(x_\alpha= \alpha x+ (1-\alpha) p_x\). Suppose \(C_p(Q)\) is the vector space \(C(Q)\) with the \(L^p\)-norm \((|x |)_p =(\int_Q |x|^p dH)^{1/p}\), where \(\mu\) denotes a positive Borel measure on \(Q\) and \(L^1(\Omega,\mu)\) is the real Banach space of all \(\mu\)-integrable functions (equivalence classes) on \(\Omega\), where \((\Omega,\mu)\) is a positive measure space. The paper consists of three sections including section 1 of the introduction. In the second section optional sunny selections of metric projections onto balls are determined for \(C_p(Q)\) \((1\leq p\leq\infty)\) and their optional Lipschitz constants are computed. Moreover, the uniqueness of the optimal sunny selection is proved for the Banach space \(C(Q)\). Main results: (1) \(K_{\mathcal P} (C(Q)) =1\) (ii) A sunny optimal selection \(P\) of the metric projection \({\mathcal P}: C(Q) \to 2^{B-\infty}\) is unique. In the third section optimal sunny selections of metric projections onto balls are determined for \(L^1 (\Omega,K)\), and their optimal Lipschitz constants are computed. Main results: (i) \(K_{\mathcal P} (L^1(\Omega, \mu))= 2\) if \(L^1(\Omega,\mu)\) is infinite dimensional (ii) \(K_{\mathcal P} (L^2(\Omega,\mu)) <2\) if and only if \(L^2(\Omega,\mu)\) is a finite-dimensional space.
scientific article; zbMATH DE number 803706