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 [ ].