AMBIGUOUS LOCI OF THE METRIC PROJECTION ONTO COMPACT STAR-SHAPED SETSIN A BANACH-SPACE

Let E be a real Banach space and I(E) the family of all nonempty compa ct starshaped subsets of E. Under the Hausdorff distance, I(E) is a co mplete metric space. The elements of the complement of a first Baire c ategory subset of I(E) are called typical elements of I(E). For X is a n element of I(E) we denote by pi(X) the metrical projection onto X, i .e. the mapping which associates to each a is an element of E the set of all points in X closest to a. In this note we prove that, if E is s trictly convex and separable with dim E greater than or equal to 2, th en for a typical X is an element of I(E) the map pi(X) is not single v alued at a dense set of points. Moreover, we show that a typical eleme nt of I(E) has kernel consisting of one point and set of directions de nse in the unit sphere of E.