On Closures of Preimages of Metric Projection Mappings in Hilbert Spaces

The closure of preimages (inverse images) of metric projection mappings to a given set in a Hilbert space are investigated. In particular, some properties of fibers over singletons (level sets or preimages of singletons) of the metric projection are provided. One of them, a sufficient condition for the convergence of minimizing sequence for a giving point, ensures the convergence of a subsequence of minimizing points, thus the limit of the subsequence belongs to the image of the metric projection. Several examples preserving this sufficient condition are provided. It is also shown that the set of points for which the sufficient condition can be applied is dense in the boundary of the preimage of each set from a large class of subsets of the Hilbert space. As an application of obtained properties of preimages we show that if the complement of a nonconvex set is a countable union of preimages of convex closed sets then there is a point such that the value of the metric projection mapping is not a singleton. It is also shown that the Klee result, stating that only convex closed sets can be weakly closed Chebyshev sets, can be obtained for locally weakly closed sets.


Introduction
Let (X, · ) be a real normed vector space. For every nonempty subset S ⊂ X the distance function from the subset S is denoted by d S (·), that is, For a given subset S ⊂ X we put and define the fiber over s as Of course D S (S) = D s (S), whenever S = {s}. Moreover, it follows from the continuity of d S (·) and d S (·) that D S (S) is closed.
The metric projection mapping on S is defined by where "cl " stands for the strong topological closure. We recall that a set S is said to be Chebyshev if P S (x) reduces to exactly one element for all x ∈ X.
Let us denote the family of weakly closed subsets of cl weak S as follows C ∈ W (S) ⇐⇒ C is weakly closed and C ⊂ cl weak S, where "cl weak " stands for the weak closure (the closure with respect to the weak topology). Let us also observe that that for every x ∈ X, s ∈ cl S we have  (1) and D (S ) (S) = {x ∈ X | ∃s ∈ S : x ∈ D s (S)} is related to the mapping defined in (2). However, if S ∈ W (S), S ⊂ cl S, and X is a reflexive Banach space, then D S (S) = D (S ) (S). Moreover assuming that S is a closed subset of cl S such that cl D S (S) = cl int D S (S), we have D S (S) = cl D (S ) (S), whenever P S (·) = ∅ on a dense subset of a Banach space (for example in the case of a reflexive locally uniformly convex space, see [28,Theorem 5]). The equalities D S (S) = D (S ) (S) or D S (S) = cl D (S ) (S), which are valid in several important cases, allows us to investigate the preimages of the metric projection on S using sets D S (S) instead of P −1 S (S ) in several important cases. In particular, if X is a real Hilbert space then D S (S) = cl P −1 S (S ) for every closed subset S of cl S such that cl D S (S) = cl int D S (S). That is the closure of the preimage of a subset S of cl S for the metric projection mapping to a given set S is the set given in (1), whenever cl D S (S) = cl int D S (S).
One of the primary questions concerning the best approximation theory is the question about the convergence of a minimizing sequence. In the language of preimages this question can be expressed in the following way: assume that u ∈ cl S and {s i } i∈N is a sequence of elements of S such that u ∈ D {s 1 ,s 2 ,... } (S) and u ∈ i∈N D s i (S), under which conditions the sequence contains a convergent subsequence? In Section 3 sufficient conditions for the convergence of a minimizing sequence are given, whenever u belongs to the boundary of D S (S) and S has some properties, see Proposition 3.1. It is also checked that desired properties of S can be guaranteed in several cases (they are listed in remarks in this section). As a simple consequence of the sufficient condition it is obtained that any "smooth" point u from the boundary of D s (S) has the property that any sequence of "almost" nearest points from S has to converge to the nearest point to u, whenever the interior of the fiber is nonempty. In Section 4 results showing the density of "smooth points" in boundaries of some sets are presented. The nonemptiness of a fiber ensures that if the inverse mapping to the projection on S is injective (i.e. the mapping d ⇒ D d (S), d ∈ S is such that D d 1 (S) ∩ D d 2 (S) = ∅, d 1 = d 2 , d 1 , d 2 ∈ S), then s is a supporting point of S, see Corollary 5.2 in Section 5. It is also observed that if the inverse mapping to the projection on S is injective then S has no isolated point, see Corollary 5.4 . In Section 7 a result characterizing those cases whenever a converging subsequece does not exist is provided, see Theorem 7.1. In Section 2 basic information is gathered. Moreover, some basic properties of fibers can be found there, for example like: convexity and closedness, see Lemma 2.3; the distance function is Fréchet differentiable on int D s (S), this is a direct consequence of Lemma 2.4.
Properties of the metric projection mapping are useful in investigating sets and differential properties of the distance function from a set. For example, it is known that if the metric projection of a Chebyshev set is continuous then the set is convex, see [1, Corollary page 237] for details in Hilbert spaces and to [34,Theorem 4] in smooth reflexive spaces. The continuity of the metric projection mapping is closely related to the differentiability of the distance function from a set, see [35, page 56] and also [33,34] for more on the link between the differentablity of the distance function and the continuity of the metric projection mapping. Several aspects of differentiability of the squared distance or the continuity of the metric projection can be also found in [1-3, 5, 7, 11, 13, 14, 18, 19, 23, 30, 36]. Herein we use the L.P. Vlasov condition given in Section 6, see (22), to preserve the continuity of the metric projection in the case S is locally weakly closed or the complement of S is included in a countable union of preimages of convex closed sets, see Theorems 8.7 and 8.10 in Section 8. Thus the convexity of S is preserved in these cases.
Let us recall that the convexity of Chebyshev sets in Hilbert space is regarded as one of the most important problems of abstract approximation theory: see [24], where it was posed: "However, even in a Hilbert space it remains unknown whether a Chebyshev set must be convex, or, equivalently, whether it must be weakly closed"; see [22,Problem 5] The Possible Convexity of a Chebyshev Set in a Hilbert Space, where a recent review of some achievements in solving the problem is given.
The author would like to thank Prof. M. Turzański for several stimulating discussions during writing of this paper.

Preliminaries
In this section some properties of subsets of Hilbert space are gathered.
For every real r > 0 and every x ∈ X we denote by B X (x, r) (resp. B X [x, r]) the open (resp. closed) ball centered at x and of radius r, the unit sphere of X is denoted by S X [0, 1] := {x ∈ X | x = 1}, the boundary of a subset D ⊂ X is denoted by fr D, where fr D := cl D \ int D, and "int " stands for the interior of D. For given x, y ∈ X we put [x, y] := {tx + (1 − t)y|t ∈ [0, 1]} and ]x, The topological dual space of X is denoted by X * , its dual norm by * , that is, where ·, · is the duality pairing between X and X * . If X is a Hilbert space with a real inner product (a real Hilbert space), then x, y = 1 4 ( x + y|| 2 − x − y 2 ) see for example [12, page 25, (1.9)]. When there is no risk of confusion, we will write x * in place of x * * . The closed unit ball centered at the origin of X * (resp. X) is denoted by B X * (resp. B X ).
In the Lemma below it is recalled that the distance function from a set S is a concave function, whenever the complement of the set is convex, see for example [20,Proposition 1,p. 66] and comments following the Proposition. Lemma 2.1 Let X be a Banach space, U ⊂ X be a convex open set, S := X \ U , and z ∈ U . Then for every x, y ∈ U, α > 0, β > 0 such that α + β = 1, z = αx + βy we have Below the convexity of sets D s (S) is shown, whenever X is a real Hilbert space. Let us start with a simple property of the norm.

Lemma 2.2
Let H be a Hilbert space, and let s, u, x, y ∈ H, α > 0, β > 0 be given such Proof For x, y, s, u ∈ H such that the inequalities in (6) are valid we have

Lemma 2.3
Let H be a real Hilbert space, S ⊂ H and s ∈ cl S be given. The set D s (S) is convex and closed.
Proof of the convexity: Let x, y ∈ D s (S), α > 0, β > 0, α + β = 1, z = αx + βy, u ∈ S. If then it follows from Lemma 2.2 that of closedness: Let {d i } i∈N be a sequence of elements of subset D s (S) such, that Let us assume thats ∈ cl S, u ∈ Ds (S) \ {s}. It is of interest that each point u ρ ∈ Ds(S) has the following property where u ρ := Take a sequence {μ i } i∈N such that μ i ∈]0, 1[ for all i ∈ N and μ i ↓ 0, and a sequence {s i } i∈N elements of S satisfying so we obtain 2 u −s,s − s i + s − s i 2 ≥ 0 and consequently hence 0 ≥ 1 − ρ, which is impossible.

Corollary 2.5
Let H be a real Hilbert space, S ⊂ H be a nonempty subset and S ⊂ cl S be a given closed nonempty subset such that int D S (S) = ∅; and u ∈ D S (S) Observe that for ρ := 1 1+t we have If a directional derivative of the distance function is equal to 1 for a unit vector at some point outside the considered set, then any minimizing sequence is converging, this is a simple consequence of [15,Proposition 2.3]. Below it is shown that the diameter of the set of minimizing points is related to the value of the directional derivative of the distance function from the set.

Lemma 2.6
Let H be a real Hilbert space, and S ⊂ H be a nonempty subset, u ∈ cl S, h ∈ S H [0, 1] be given. If Proof Suppose that Take any for every i ∈ N. Observe that for every i ∈ N we have and consequently The condition lim t↓0 for i ∈ N large enough. Hence for i ∈ N large enough, so which by the choice of δ and is impossible, thus (8) is valid.
In fact we have equality in (8). The reverse inequality is obtained in Lemma 8.1. Let us also mention that we have the strong convergence of any minimizing sequence, whenever γ = 1.
It is known that norms of elements of a weakly converging sequence can be far from the norm of its weak limit. This is a drawback of the weak convergence which sometimes can be overcome, for example whenever the Kadec-Klee property is valid. Below a condition allowing to estimate from below the norm of the weak limit of a sequence by norms of its elements is provided.

Lemma 2.7
Let H be a real Hilbert space, X, Y ⊂ H be closed subspaces such that H = X + Y , X has a finite dimension and x, y = 0 for every x ∈ X, y ∈ Y ; and > 0 be given. Then for all sequences where " weak −→" stands for the weak convergence.
Proof Assume that lim sup i−→∞ x i = lim i−→∞ x i , if not then a proper subsequence can be chosen. Observe that by the local compactness of X we can choose a subsequence such that {P X (x i k )} k∈N is converging, again, without loss of generality we may assume that {P X (x i )} i∈N converges. We have We finish this section with an observation, which is intuitively obvious, namely if D is a set, u ∈ fr D is a "smooth" boundary point, then moving from u along a tangent direction to D we are sufficiently close to the set D. For the sake of the reader's convenience we provide this property in the Hilbert space setting.

Lemma 2.8 Let H be a real Hilbert space with, D ⊂ H be a subset with nonempty interior. Assume that for
. For every z ∈ S H [0, 1] for which z, u * = 0 and every t ∈ R we have

Points from the Set of Points of Approximate Compactness
Let S ⊂ H and u ∈ S be given. We say that u is a point of approximate compactness of S if each minimizing sequence, that is s i ∈ S and u − s i −→ d S (u), contains a convergent subsequence to the nearest point to u. The set of the all points of approximate compactness of S we denote AC(S), we refer to [2, page 1131] for more information on the notion. Following [2, see page 1131] we say that S is approximately compact whenever H \ S = AC(S), see [2] for historical references. It was observed in (7) that for each u ∈ fr D s (S), ρ ∈ [0, 1[ and each sequence {s i } i∈N of elements of S such that ρu has to converge to s, so ρu + (1 − ρ)s ∈ AC(S). It is also interesting that each "smooth" point u ∈ fr D s (S) is an element of AC(S), whenever int D s (S) = ∅, this is a consequence of the result below. In fact, in Proposition 3.1 a more general characterization of some points from AC(S) is given. This characterization allows us to preserve the continuity of the metric projection mapping at those "smooth" points for which the metric projection is a singleton. Of course, the question is when the assumptions of Proposition 3.1 are fulfilled. In Remarks 3.2, 3.3, 3.4, 3.5 some results answering the question are presented.
Assume that for a given sequence We may assume that for some δ > 0 and all i ∈ N we have For i large enough have also Since u ∈ AC(S ) we are able to find a subsequence of the sequence {s i } i∈N which is convergent, says i k −→s ∈ cl S ∩ P S (u) (keep in mind the assumption P S (u) = {s}), which impliess i k −→s and The compactness of the segment [−Mu * , Mu * ] ensures the existence of a convergent sub- In order to show that assumptions of Proposition 3.1 are not difficult to verify we present several examples below. In the first example S is assumed to be convex. The convexity of S gives the equality H \ S = AC(S ), so S is aproximately compact.

Remark 3.2
Let H be a real Hilbert space, S ⊂ H be a nonempty subset and S ⊂ cl S be a nonempty convex closed subset. Assume that for a given u ∈ S we havē and there is u * ∈ S H [0, 1] such that Put Thus the assumptions of Proposition 3.1 are satisfied.
Proof Take any z ∈ S H [0, 1] satisfying z, u * = 0 and fix i ∈ N. Because of (10) we are Additionally P S (u) = {s}, since S is closed and convex.
Approximately compact sets have the property that any Chebyshev set which is approximately compact has to be convex, see [2]. In Section 8 a local version of this result is presented, see (11) and Theorem 8.7 below.

Remark 3.3
Let H be a real Hilbert space, S ⊂ H be a nonempty subset, u ∈ S, S ⊂ cl S be given ands ∈ S . Assume that for every sequence {s i } i∈N in S , and that there is Put Thus the assumptions of Proposition 3.1 are satisfied.
Proof In order to show that u ∈ AC(S ) let us take a minimizing sequence . It is known that from each minimizing sequence we are able to choose a weakly convergent subsequence, see [12,Theorem 3.139 property of the norm in the Hilbert space setting is used to get the strong convergence). Observe also that we have the inclusion P S (u) ⊂ P S (u), which by (11) givess Take any z ∈ S H [0, 1] satisfying z, u * = 0 and fix i ∈ N. Because of (12) we are able It is known that P S (u) is a singleton, whenever the distance function d S (·) is Fréchet differentiable at u ∈ cl S, see also [13,Theorem 2.4], a part of the proof of the Remark below can be also used to get it. We use the idea behind the result to get a local sequential weak closedness of S, that is we prove that (11) holds true, whenever d S (·) is Fréchet differentiable at u ∈ cl S. Remark 3.4 Let H be a real Hilbert space, S ⊂ H be a nonempty subset, u ∈ S, S ⊂ cl S be given ands ∈ S . Suppose that the Fréchet derivative of the distance function d S (·) exists at u, that is for every sequence Thus the assumptions of Proposition 3.1 are satisfied.
Proof It follows from Lemma 2.6 that P S (u) = ∅, it is enough to apply Lemma 2.6 for h := x * and observe that h = 1, see [ , then the strong convergence s i −→s is again a consequence of Lemma 2,6.
Observe also that we have the inclusion P S (u) ⊂ P S (u), which implies then take any z ∈ S H [0, 1] satisfying z, u * = 0. As in the proof of Remark 3.3 we are able Then {s} = P S (u) ∩ S and for every sequence Thus the assumptions of Proposition 3.1 are satisfied.
Proof Put then it follows from Lemma 2.6 that condition (15) holds true, hence {s} = P S (u) ∩ S .
Let us exclude the case a contradiction. In order to finish the proof it is enough to repeat the reasoning from Remark 3.4.
There are two natural examples for S to fulfill condition (14), namely if S ⊂ a + {h ∈ H | x * , h = 0} for some a ∈ H, x * ∈ S H [0, 1] and u −s = αx * for some α > 0, then it is obvious that (14) is satisfied. Condition (14) is also satisfied, whenever S is convex and

Density Properties
Let S, S be a subsets of a real Hilbert space. "Smooth points" of the boundary of D S (S) have the "convergence property" as it is observed in the previous section. Of course the state of the art in variational analysis allows us to expect that "smooth points" of the boundary are dense in the boundary. It seems that using ideas from [27], see also [38], we can get this kind of density in every Banach space with the norm Fréchet differentiable and having the Kadec-Klee property. Since we are interested only in the Hilbert space setting we provide such a result in a Hilbert space. The key tool to get it is the Borwein-Preiss Variational Principle, we use a form of the Variational Principle from [6, Theorem 2.5.2]. (X, d) be a complete metric space. We say that a continuous function ρ : X × X → [0, +∞] is a gauge-type function provided that (i) ρ(x, x) = 0 for all x ∈ X, (ii) for any ε > 0 there exists δ > 0 such that ρ(y, z) ≤ δ implies d(y, z) < ε for all y, z ∈ X.

Theorem 4.2 Let (X, d) be a complete metric space and let f : X → R ∪ {+∞} be a lsc function bounded from below. Suppose that ρ is a gauge-type function and (δ i ) ∞
i=0 is a sequence of positive numbers, and suppose that > 0 and x 0 ∈ X satisfy There exist x ∈ X and a sequence ( Below we present a result which is a consequence of [28,Theorem 4]. It says that for every given point u ∈ S there is a point u close to the given point such that P S (u ) = ∅. For the sake of the reader's convenience a proof is provided.
, so x ∈ U , and there are x * ∈ H and M > 0 (apply (iii)) such that which forces weak −→" stands for the weak convergence. By (16) we have for a proper choice of t i > 0, t i −→ 0 and i large enough, hence It follows from (18) and the weak lower-semicontinuity of x − · that It is easy to observe that s ∈ K ∩ cl U , so s ∈ cl U \ U .
Let us recall that x ∈ B H [w, δ 16 ], which implies the first part of statement. The choice of a convergent subsequence, which is demanded in the second part of the statement, is guaranteed by the above reasoning. In order to see it, let us fix a sequence {w i } i∈N in K such that w i − x −→ d K (x). Now take any weakly convergent subsequence {w i n } n∈N ⊂ {w i } i∈N . Repeating the reasoning, which is presented above to get the strong convergence of the sequence {s i } i∈N , we get the strong convergence of {w i n } n∈N to s (keep in mind that A convex set has the property that the closure of its interior recovers the closure of the set, whenever the interior is nonempty. It is interesting that the set of supporting points is dense in the boundary of a convex set with nonempty interior, this result was due to Bishop-Phelps and it was an answer to an open problem by V. Klee. Of course the class of sets having this property is larger than the class of convex sets with nonempty interior. Below it is shown that the set of points from the boundary of some set, belonging to the boundary of a ball contained in the set, is dense in the boundary of the set, whenever the considered set belongs to the class. then it has a converging subsequence (with respect to the norm topology), say , which implies the statement. Below we provide another example of a possibly non-convex set having the following property cl D = cl int D. If a set has the property above, then it follows from Proposition 4.5 that there is a dense subset of fr D, say F ⊂ fr D, such that for every u ∈ F there is x ∈ int D for which d H\int D (u) = u − x .
then u ∈ cl int D S (S).
If there is a sequence {u i } i∈N in H \ D S (S) such that u i −→ u ρ , then for every i ∈ N put There is a sequence for i ∈ N large enough, which contradicts the choice of the sequence {s i } i∈N . Thus u ρ ∈ int D S (S) and consequently u ∈ cl int D S (S).

Sets for which there are Points at which the Metric Projection is not a Singleton
In this section several results ensuring, for a giving set, the existence of points at which the metric projection is not a singleton are presented.

Corollary 5.1 Let H be a real Hilbert space, S ⊂ H be a nonempty subset and S ⊂ cl S be a given closed convex nonempty subset such that int D S (S) = ∅; and u ∈ cl int D S (S) \ int D S (S), u ∈ S , s ∈ S ∩ P S (u), u + t (u − s) ∈ D s (S) for every t ∈]0, ∞[. If there is x ∈ int D S (S) for which d H\int D S (S) (x) = u − x and u − s, u − x = 0, then there exists d ∈ cl S, d = s such that u ∈ D d (S).
Proof Let us notice that if u + t 0 (u − s) ∈ D S (S) for some t 0 ∈]0, ∞[, then u + 2 −1 t 0 (u − s) ∈ D s (S), since the convexity of S , but this contradicts to u + t (u − s) ∈ D s (S) for every t ∈]0, ∞[. Hence, there are sequences A simple consequence of the Corollary above is A direct consequence of Corollary 5.2 is that the existence of nonsmooth points in the boundary of a set S can cause that images of the projection mapping P S (·) are not singletons.

Corollary 5.3 Let H be a real Hilbert space, S ⊂ H be a closed nonempty subset ands
We have also for every i ∈ N. It follows from (21) that s i ∈ S \S for every i ∈ N. It is easy to observe that weakly closed subsets are approximately compact, thus using Lemma 2.8 and Proposition 3.1 we infer that the sequence {s i } i∈N has a convergent subsequence, say s i k −→ d. If d = s, then using (20) we obtain s i k ∈ S for k ∈ N large enough, but this is impossible in view of (21). Of course u ∈ D d (S), and consequently u ∈ D d (S) ∩ D S (S).

Corollary 5.5 Let H be a real Hilbert space, S ⊂ H be a closed nonempty subset, and S ⊂ S be a closed convex nonempty subset such that int D S (S) \ S = ∅ and cl int D S (S) \ (int D S (S) ∪ S ) = ∅. Then
Proof It follows from Proposition 4.5 that for every w ∈ cl int D S (S)\int D S (S) and every (u ). It follows from Lemma 2.4 that the sequence {s i } i∈N is convergent, say to d ∈ cl (S \S ), so we are done. Now let us consider the case u +t (u −P S (u )) ∈ D P S (u ) (S) for every t > 0 and apply Corollary 5.1 to get the statement. For this reason let us observe that if P S (u ) ∈ cl (S \ S ), then we are done, however if P S (u ) ∈ cl (S \ S ) then u − x , u − P S (u ) > 0, and all assumptions of Corollary 5.1 are fulfilled, so the statement is a consequence of Corollary 5.1.

The Vlasov Condition
In this section we present some results related to the Vlasov condition, see (6.1) below. In order to characterize the role of the Vlasov condition let us start with the following result which was obtained by L. P. Vlasov [34,Theorem 3] in the setting of a Banach space (X, · ) whose dual norm · * of X * was assumed to be strictly convex. It states that in some Banach spaces the continuity of the metric projection on a set S preserves the convexity of the set. A new proof of this result for X being a Banach space whose norm is uniformly Gâteaux differentiable is given in [23,Theorem 4.4]. In the next section the Vlasov condition is applied to get convexity of Chebyshev sets in some new cases.

Theorem 6.1 Let X be a Banach space whose norm is uniformly Gâteaux differentiable and S ⊂ X be a Chebyshev set with continuous metric projection. Then S is convex.
The continuity of the metric projection can be also preserved by checking if the Vlasov condition is satisfied, see [35, page 56] and also [33,34] since x − s * ≤ d S (x). Observe that (24) implies (22), so it is enough to get inequality (25) in order to finish the proof. Suppose the contrary, that is for some > 0. By (23) and (26) we have a contradiction.

Tangent Cones to Preimages of the Metric Projection on a Set of Isolated Points
Let S ⊂ H be a subset of isolated points, u ∈ S and {s} = P S (u). In this section the tangent cone to P −1 S (s) is calculated, whenevers is not a weak cluster point of S. Let us recall that for every nonempty subset A ⊂ H and a ∈ cl A by we denote the tangent cone (Bouligand tangent cone or the contingent cone, see [ The choice of the sequences is preserved by Proposition 4.5 and Lemma 2.8. It follows from Proposition 3.1 that either there is a converging subsequence of the sequence {s i } i∈N (but this is impossible) or for each i there is k(i) ∈ N such that (keep in mind that u i ∈ fr Ds(S) is a "smooth" point) Because of assumption 2, there is a subsequence {i n } n∈N such that k(i n ) < k(i n+1 ). Put y n := u i n and d n := s k(i n ) . For every n ∈ N we have y n −s 2 = y n − d n 2 (28) and y n − d n 2 = y n − u + u − d n 2 = y n − u 2 + 2 y n − u, u − d n + u − d n 2 ≥ y n − u 2 + 2 y n − u, u − d n + u −s 2 = y n − u 2 + 2 y n − u, u − d n + u − y n 2 + 2 u − y n , y n −s + y n −s 2 , thus by (28) we get 0 ≥ u − y n + u − y n −1 (y n − u), u − d n − y n +s .
In order to finish the proof let us observe that for all t > 0, z ∈ T (Ds(S), u) and i ∈ N we have Thus assumptions from 1. to 3. ensure that for every z ∈ T (Ds(S), u), so we have the inclusion Observe that if z, s * −s < 0, then, by assumption 3., s − u,s − s * > 0, so using (32)

Some Sufficient Conditions for the Convexity of Chebyshev Sets
The problem of convexity of Chebyshev sets of a Hilbert space is old, we refer to [26, K4 Farthest and Nearest Points in Hilbert Space, Comments by Grünbaum] for some historical aspects. In this section we exhibit some kinds of subsets in Hilbert space which are convex whenever they are Chebyshev. First let us recall that due to V. Klee we know that weakly closed Chebyshev sets are convex, see [24,25]. Another way to get the convexity of a Chebyshev set is to assume a differentiability of the distance function outside the set. Namely, if the distance function to a Chebyshev set is Fréchet differentiable at all points outside the set then the set is convex too, we refer to [13,14,18,19] for details. Due to L. P. Vlasov we know also that the continuity of the metric projection (which implies the convexity) can be obtained by checking if the Vlasov condition is satisfied, see (22) and [35, page 56], we refer also to [33,34]. The Vlasov result is also important in the proof of Theorem 8.10. At this moment it seems that this tool is essential in detecting whether a Chebyshev set is convex or not. In order to shed light on the meaning of the Vlasov condition let us observe that if it is violated in some special points then the set cannot be a Chebyshev set, namely

Then for all sequences {s
then the statement is obvious. Let us suppose that s ∈ S, u ∈ D s (S), and for every i ∈ N (we can assume this without loss of generality, otherwise we omit a finite number of indices), where r := u − s and of course u as well as (34) and by the last inequality in (34) we obtain so using (34) we get for i ∈ N, which implies the statement.

Corollary 8.6
Let H be a real Hilbert space, S ⊂ H be a nonempty subset ands ∈ S, u ∈ Ds (S) \ S be given such that for every sequence {s i } i∈N elements of S we have Then the implication Since the sequence {s i } i∈N is bounded we may suppose that s i weak −→s ∈ S (if not we choose a proper subsequence). Observe that and consequently s =s, u ∈ Ds(S) ∩ D s (S).
Below a generalization of the V. Klee result is provided in the Hilbert space setting, we refer to [24, whenever {s i } i∈N is in B H [s, δ] ∩ S, and consequently (36) is satisfied.
Below it is stated that if S = i∈N S i , where S i are closed convex nonempty subsets, then S can not be Chebyshev set, whenever it is not convex. By a similar reasoning as above we get Hence it follows from Theorem 6.1 that S is convex, a contradiction. By the Kuratowski-Zorn Lemma we find I ⊂ N such that j 1 = j 2 , whenever j 1 , j 2 ∈ I ; and int D S j (S) ∩ (H \ S) = ∅ and ∀x ∈ int D S j (S) \ S j , t∈]0,∞[ B[P s j (x ) + t (x − P s j (x )), t x − P s j (x ) ] ∩ S = {P s j (x )} for every j ∈ N, and int H \ S ∪ s∈I int D S i (S) = ∅. It follows from Corollary 6.2 that (22) holds true for all x ∈ S, and by the L.P. Vlasov results the set S is convex, but this is impossible. Theorem 8.10 when compared with [2, Theorem 2.19] has the following differences: first, it is given in a Hilbert space, while [2,Theorem2.19] is given in a more general space, namely in the smooth Efimov-Stechkin space; second, it is not assumed that its boundary is included in a countable union of hyperplanes, as it was done in [2,Theorem]. So, it is natural to expect a result combining advantages of both theorems, but this is not the aim of this paper.
Let us recall the following problem raised by K. Goebel and R. Schöneberg: Does there exist a convex body Y ⊂ H such that the boundary of Y is a Chebyshev set with respect to its convex closure, in other words is S := H \ int Y a Chebyshev set for some convex bounded set Y ⊂ H having nonempty interior, see [17, Problem 1, page 466]? A. P. Bosznay gave the answer in the negative the question, whenever the boundary of Y is included in a countable union of hyperplanes, see [4,Theorem,page 143], see also [2,Theorem 2.19], where a generalization of this result was given and several results on the convexity of Chebyshev sets can be also found. In view of Corollary 8.11 it is enough to know that a part of the boundary is flat in order to answer the question in the negative, the details are presented in the Corollaries below. First, is shown that condition (14) can be used to have S in some halfspace.

Corollary 8.11
Let H be a real Hilbert space, S ⊂ H be a nonempty closed subset,s ∈ S, u ∈ S, > 0 be given such that d S  However, the inclusion above is not valid, since it contradicts to (41). Indeed, if u i −→ u, d i ∈ cl (S \ S ), u i ∈ D d i (S) ∩ D S (S) for every i ∈ N, then by the inclusion in (41) we get d i ∈ S for every i ∈ N. Observe that d i ∈ cl (S \ S ) and d i ∈ S imply d i −s = , which contradicts the continuity of P S , that is we should have had d i = P S (u i ) −→s since u i −→ u. Thus u + t (u −s) ∈ int D S (S) for every t > 0, otherwise we repeat the reasoning above to get a contradiction with ut := u +t(u −s), wheret > 0 is such that u +t(u −s) ∈ cl int D S (S) \ int D S (S). The condition u + t (u −s) ∈ int D S (S) for every t > 0, implies (43).