A remark on the lower semi-continuity of the set-valued metric projection

Abstract

where E is a normed linear space and G C E a proximial linear subspace. In this context, lower and upper semi-continuity and continuous selections are three of the most important notions: Let F: E -+ 2G be a set-valued mapping (E, G Hausdorff-spaces.) F is called lower semi-continuous (l.s.c.), iff, for each x E E and each open subset UC G with U r F(x) # o , there exists a neighborhood V C E of x such that U n F(y) f o for each y E V. F is called upper semi-continuous (u.s.c.), iff, for each x E E and each open subset UC G with F(x) C U, there exists a neighborhood V C E of x with F(y) C U for each y E V. A single-valued mapping S: E + G is called a selection for F, iff s(x) E F(x) for each x E E. A well-known theorem of Michael [4, Theorem 3.2”] states that an 1.s.c. set-valued mapping from a paracompact Hausdorff-space into a Banachspace with closed convex images always admits a continuous selection. In his paper [5], Ntirnberger introduces the following notion: Let E be a normed linear space and G C E a proximinal linear subspace. A selection S: E + G for PG is said to have the “Nulleigenschaft,” iff S(X) = 0 for each x E E with 0 E PC(x). Using this notion, the lower semi-continuity of the set-valued metric projection can be characterized as follows: 83 0021-9045/80/010083-04$02.00/0