Abstract
LetE be a real Banach space andL(E) the family of all nonempty compact starshaped subsets ofE. Under the Hausdorff distance,L(E) is a complete metric space. The elements of the complement of a first Baire category subset ofL(E) are called typical elements ofL(E). ForX∈L(E) we denote byπ χ the metrical projection ontoX, i.e. the mapping which associates to eacha∈E the set of all points inX closest toa. In this note we prove that, ifE is strictly convex and separable with dimE≥2, then for a typicalX∈L(E) the mapπ χ is not single valued at a dense set of points. Moreover, we show that a typical element ofL(E) has kernel consisting of one point and set of directions dense in the unit sphere ofE.
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De Blasi, F.S., Kenderov, P.S. & Myjak, J. Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space. Monatshefte für Mathematik 119, 23–36 (1995). https://doi.org/10.1007/BF01292766
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DOI: https://doi.org/10.1007/BF01292766