Metric projection onto convex sets (Q1066391)

Let X be a real linear normed space and let \(X^*\) be its conjugate space. For any nonempty sets M, N in X, define \(d(x,M)=\inf \{\| x- y\|:\quad y\in M\}\) and \(d(M,N)=\sup \{d(x,N):\quad x\in M\}.\) The set-valued mapping \(P_ Mx=\{y\in M:\quad \| x-y\| =d(x,M)\}\) is called the metric projection from X onto M. A set M in X is called a strict convex subset, if M is convex, closed, has nonempty interior, and its boundary contains no interval. A real linear normed space X is called (RB\({\mathbb{R}})\) space (or \(X\in (RBR))\), if for every \(f\in X^*\), \(\| f\| =1\), \(\Gamma_ f=\{x:f(x)=\| x\| =1\}\) is either empty, or singleton, or a strict convex subset in the hyperplane \(\{x\in X:f(x)=1\}\). A metric projection \(P_ M\) is called lower semi-continuous if for any \(x\in X\), \(y\in P_ Mx\) and \(x_ n\to x\), there holds \(d(y,P_ Mx_ n)\to 0\) is called lower H-semi-continuous if for any \(x\in X\) and \(x_ n\to x\) there holds \(d(P_ Mx\), \(P_ Mx_ n)\to 0.\) In this paper the authors study the relations of the various continuous properties of metric projection and the structure of Banach space. The main result is the following: Theorem 4. For Banach space X, the following statements are mutually equivalent: (1) \(X\in (RBR)\); (2) for any 3-dimensional subspace \(X_ 3\), of X, \(X_ 3\in (RBR)\); (3) for any 3-dimensional subspace \(X_ 3\) of X the metric projection from X onto any closed convex subset of \(X_ 3\) is lower semi-continuous; (4) the metric projection from X onto any bounded compact convex subset \(M\subset X\) is lower semi-continuous; (5) the metric projection from X onto any bounded compact convex subset \(M\subset X\) lower H-semi-continuous.