Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space

Abstract

Abstract. Let E be a real Banach space and 5P(E) the family of all nonempty compact starshaped subsets of E. Under the Hausdorff distance, So(E) is a complete metric space. The elements of the complement of a first Baire category subset of 5g(~:) are called typical elements of 5~(~). For XeS~(E) we denote by gx the metrical projection onto X, i.e. the mapping which associates to each a~: the set of all points in X closest to a. In this note we prove that, if E is strictly convex and separable with dim ~/> 2, then for a typical X E 5~(~) the map nx is not single valued at a dense set of points. Moreover, we show that a typical element of 5~(E) has kernel consisting of one point and set of directions dense in the unit sphere of E.

On the cut locus in Alexandrov spaces and applications to convex surfaces

“…This result of [34] has already been repeatedly strengthened, for example in [3], [4], [5], [6], [41], [7]. Theorem 3.…”

“…The ambiguous locus has been investigated by -among others -de Blasi and Myjak [3], [4], [5]; de Blasi, Kenderov and Myjak [6]; Myjak and Rudnicki [21]; Zhivkov [41]; de Blasi and Zamfirescu [7].…”

Extending Stechkin′s theorem and beyond

“…Already [8,Theorem 3] is a generalization of an old result of the author [7], which had been repeatedly strengthened in various ways, for example, in [1,2,3,4,5,9].…”

Properties of typical bounded closed convex sets in Hilbert space

“…Zamfirescu [26] proved for metric projections that typically the complement of L 1 (M) is dense. Further development for either metric projections or antiprojections is to be found in, for example, [6,7,8,9,14,28,29,30,31].…”