Cardinality of the metric projection on typical compact sets in Hilbert spaces

Abstract

The metric projection mapping πX plays an important role in nonlinear approximation theory. Usually X is a closed subset of a Banach space [ ] and, for each e∈[ ], πX(e) is the set, perhaps empty, of all points in X which are nearest to e. From a classical theorem due to Stečkin [7] it is known that, when [ ] is uniformly convex, the metric projection πX(e) is single valued at each typical point e of [ ] (in the sense of the Baire categories), i.e. at each point e of a residual subset of [ ]. More recently Zamfirescu [8] has proven that, if X is a typical compact set in ℝn (in the sense of Baire categories) and n[ges ]2, then the metric projection πX(e) has cardinality at least 2 at each point e of a dense subset of ℝn. This result has been extended in several directions by Zhivkov [9, 10], who has also considered the case of the metric antiprojection mapping νX (which associates with each e∈[ ] the set νX(e), perhaps empty, of all ∈X which are farthest from e). For this mapping De Blasi [2] has shown that, if [ ] is a real separable Hilbert space with dim[ ]=+∞ and n is an arbitrary natural number not less than 2, then, for a typical compact convex set X⊂[ ], the metric antiprojection νX(e) has cardinality at least n at each point e of a dense subset of [ ]. A systematic discussion of the properties of the maps πX and νX, and additional bibliography, can be found in Singer [5, 6] and Dontchev and Zolezzi [3].In the present paper we consider some further properties of the metric projection mapping πX, with X a compact set in a real separable Hilbert space [ ]. If dim[ ]=n and 2[ges ]n<+∞, it is proven that for a typical compact set X⊂[ ], the metric projection πX(e) has cardinality exactly n+1 at each point e of a dense subset of [ ], while the set of those points e∈[ ] where πX(e) has cardinality at least n+2 is empty. Furthermore it is shown that, if dim[ ]=+∞, then for a typical compact set X⊂[ ] the metric projection πX(e) has cardinality at least n (for arbitrary n[ges ]2) at each point e of a dense subset of [ ]. Incidentally we obtain a characterization of the dimension of the space [ ] by means of a typical property holding in the space of the compact subsets of [ ].