The linear selections of metric projections in the \(L_ p\) spaces (Q1176335)

Let \(Y\) be a non-empty subset of a normed linear space \((X,\| \|)\) and \(P_ Y:X\to 2^ Y\) the metric projection (nearest point mapping) defined by \(P_ Y(x)=\{y\in Y:\| x-y\|=d(x,Y)\}\) for any \(x\in X\). If \(P_ Y(x)\neq \emptyset\) for each \(x\in X\) then the set \(Y\) is said to be proximinal. Any closed convex subset of a uniformly convex \(B\)-space is proximinal. A linear selection for \(P_ Y\) is a linear mapping \(s:X\to Y\) such that \(s(x)\in P_ Y(x)\) for each \(x\in X\). The author gives a characterization of those subspaces of \(L_ p\) whose metric projection is linear. He also characterizes the subspaces of \(L_ 1\) which are finitely codimensional and their metric projection admits a linear selection. The two main theorems in the paper are as follows: Theorem 2.2. Let \((T,\Sigma,\mu)\) be a purely atomic measure space and \(Y\) a closed subspace of \(L_ p (1<p<\infty, p\neq 2)\). Then the following statements are equivalent, (a) \(P_ Y\) is linear; (b) There exists a disjoint subset family \(\{A_ \lambda\}_{\lambda\in\Lambda}\) of \(T\) such that \(Y=[\oplus_{\lambda\in\Lambda}M_ \lambda]_ p\), where \(M_ \lambda\) is either \(L_ p(A_ \lambda)\) or a hyperplane of \(L_ p(A_ \lambda)\) for any \(\lambda\in\Lambda\). Theorem 3.2. Let \((T,\Sigma,\mu)\) be a finite measure space and \(Y\) an \(n\)-codimensional proximinal subspace of \(L_ 1(T)\). Then the following statements are equivalent, (1) \(P_ Y\) admits a linear selection \(P\) such that \(P1_ T=0\), (2) There exist measurable subsets \(A_ 1,A_ 2,\ldots,A_ n\) such that (a) \(A_ i\cap A_ j=\emptyset (i\neq j)\) and \(\bigcup^ n_{k=1}A_ k=T\). (b) \(Y=\{f\in L_ 1(T)\); \(\int_{A_ i}fd\mu=0\), \(i=1,2,\ldots,n\}\).