About the Lipschitz property of the metric projection in the Hilbert space

J. Math. Anal. Appl. 394 (2012) 545–551

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About the Lipschitz property of the metric projection in the Hilbert space Maxim V. Balashov ∗ , Maxim O. Golubev Department of Higher Mathematics, Moscow Institute of Physics and Technology, Institutskii pereulok 9, Dolgoprudny, Moscow region, 141700, Russia

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Article history: Received 29 September 2011 Available online 17 May 2012 Submitted by Aris Daniilidis

We consider the metric projection operator from the real Hilbert space onto a strongly convex set. We prove that the restriction of this operator on the complement of some neighborhood of the strongly convex set is Lipschitz continuous with the Lipschitz constant strictly less than 1. This property characterizes the class of strongly convex sets and (to a certain degree) the Hilbert space. We apply the results obtained to the question concerning the rate of convergence for the gradient projection algorithm with differentiable convex function and strongly convex set. © 2012 Elsevier Inc. All rights reserved.

Keywords: Hilbert space Distance function Metric projection Strongly convex set of radius R Supporting principle Mazur intersection property Gradient projection algorithm

1. Introduction and main notation Throughout this paper, E denotes a real Banach space, and E ∗ its dual. Let H be the real Hilbert space and Rn be n-dimensional Euclidean space. We denote by (p, x) the value of the functional p ∈ E ∗ at the point x ∈ E. Let Br (a) = {x ∈ E | ∥x − a∥ ≤ r }. We denote by ∂ A, int A and cl A the boundary, the interior and the closure of the set A, respectively. The diameter of the subset A ⊂ E is defined as diam A = supx,y∈A ∥x − y∥. The support function of the subset A ⊂ E is given by the formula s(p, A) = sup(p, a),

∀p ∈ E ∗ .

a∈A

We denote the normal cone to the closed convex subset A ⊂ E at the point a ∈ A by N (A, a), i.e. N (A, a) = {p ∈ E ∗ | (p, a) ≥ s(p, A)}. For any closed convex subset A ⊂ E and any vector p ∈ E ∗ we define the set A(p) = {x ∈ A | (p, x) = s(p, A)}. The distance function from the point x ∈ E to the subset A ⊂ E is given by the formula

ϱ(x, A) = inf ∥x − a∥, a∈A

and the metric projection of the point x ∈ E on the set A is defined as follows: PA x = {a ∈ A | ∥x − a∥ = ϱ(x, A)}. For a subset A ⊂ E let U (A, ϱ) be the open ϱ-neighborhood of A, i.e. U (A, ϱ) = {x ∈ E | ϱ(x, A) < ϱ}.

∗

Corresponding author. E-mail addresses: [email protected] (M.V. Balashov), [email protected] (M.O. Golubev).

0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.05.024

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It is well known that the set PA x is a singleton for any closed convex subset A ⊂ H and for any point x ∈ H . Moreover, for any pair of points x0 , x1 ∈ H we have

∥a0 − a1 ∥ ≤ 1 · ∥x0 − x1 ∥,

ai = P A x i ,

i = 0, 1.

(1.1)

The Lipschitz constant 1 in Formula (1.1) is the best possible in the general case and it is attained on the closed affine subspaces. On the other hand, consider A = BR (a) ⊂ H and any points x0 , x1 ∈ H with ϱ(x0 , A) = ϱ0 > 0, ϱ(x1 , A) = ϱ1 > 0. By the cosine theorem for the triangle x0 x1 a (with sides ∥a − x0 ∥ = R + ϱ0 , ∥x1 − a∥ = R + ϱ1 ) we easily conclude that

∥ a0 − a1 ∥ = 

R

(R + ϱ0 )(R + ϱ1 )

·



∥x0 − x1 ∥2 − (ϱ0 − ϱ1 )2 .

(1.2)

Here ai = PA xi , i = 0, 1. Thus, if there exists a number ϱ > 0 such that ϱ(xi , BR (a)) ≥ ϱ, i = 0, 1, then

∥ a0 − a1 ∥ ≤

R R+ϱ

· ∥x0 − x1 ∥.

(1.3)

The aim of the present article is to characterize all closed convex subsets A ⊂ H with the following property: for any number ϱ > 0 there exists a number C ∈ (0, 1) such that for any pair of points x0 , x1 ∈ H \ U (A, ϱ) the following inequality holds:

∥a0 − a1 ∥ ≤ C · ∥x0 − x1 ∥,

ai = P A x i ,

i = 0, 1.

(1.4)

Some results concerning estimates of the Lipschitz constant for the metric projection operator can be found in [1]. Our key idea is to use the concept of a strongly convex set of radius R. Definition 1.1 ([2–5]). A nonempty subset A ⊂ E is called strongly convex of radius R >0 if it can be represented as the intersection of closed balls of radius R > 0, i.e. there exists a subset X ⊂ E such that A = x∈X BR (x). Strongly convex sets of radius R are closely related to the classical notions of diametrically maximal sets and constant width sets; see [6–11]. They are also useful in different problems of optimization and optimal control [3,12,4,5,13]. Additional information concerning strongly convex sets of radius R > 0 can be found in the papers [2,14,15,3,4,11,5,13]. Proposition 1.1 (Supporting Principle [2,3]). A closed convex subset A ⊂ H is strongly convex of radius R if and only if for any unit vector p ∈ H and for the point {a(p)} = A(p) the following inclusion holds: A ⊂ BR (a(p) − Rp) . Proposition 1.2 ([3]; see also [11, Theorem 4.3.2]). A closed convex subset A ⊂ H is strongly convex of radius R if and only if for any unit vectors p, q ∈ H and for the points {a(p)} = A(p), {a(q)} = A(q), the following inequality holds:

∥a(p) − a(q)∥ ≤ R∥p − q∥. 2. Metric projection and strong convexity of radius R Lemma 2.1. Let A ⊂ H be a closed convex subset. Let ϱ > 0, and C ∈ (0, 1). Suppose that Formula (1.4) holds true for any x0 , x1 ∈ H \ U (A, ϱ) and ai = PA xi , i = 0, 1. Then the set A is bounded. Proof. By Danford and Schwartz [16, Corollary 12, Section 9, Chapter 5] there exists a dense subset S ⊂ ∂ A such that any point a ∈ S satisfies the condition a = PA xa for some xa ∈ H \ A. Moreover, l(a, xa ) = {a + λ(xa − a) | λ ≥ 0} ⊂ a + N (A, a). Thus PA l(a, xa ) = {a}. Fix any points a, b ∈ S. There exist points za ∈ l(a, xa ) and zb ∈ l(b, xb ) satisfying ∥za − a∥ = ∥zb − b∥ = ϱ. Then

∥a − b∥ ≤ C ∥za − zb ∥ ≤ C (∥za − a∥ + ∥a − b∥ + ∥zb − b∥) = C (2ϱ + ∥a − b∥), 2C ϱ

2C ϱ

and, therefore, ∥a − b∥ ≤ 1−C . The property cl S = ∂ A guarantees that diam A ≤ 1−C .



Due to Lemma 2.1, we can further consider only bounded sets. Theorem 2.1. Let A ⊂ H be a closed convex bounded subset. Let ϱ > 0, and C ∈ (0, 1). Suppose that Formula (1.4) holds true Cϱ for any points x0 , x1 ∈ H \ U (A, ϱ) and ai = PA xi , i = 0, 1. Then the set A is strongly convex of radius R = 1−C .

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547

Proof. Let x0 , x1 ∈ ∂ (H \ U (A, ϱ)), i.e. ϱ(xi , A) = ϱ, i = 0, 1. We have

   x 0 − a0 x1 − a1   , ∥a0 − a1 ∥ ≤ C ∥a0 − a1 ∥ + C ϱ ·  − ϱ ϱ     x0 − a0 x 1 − a1  ,  − ∥ a0 − a1 ∥ ≤ R ·  ϱ ϱ 

(2.5)

and i ϱ i ∈ N (A, ai ), i = 0, 1. Let us show that x −a



x − PA x

ϱ

 | x ∈ ∂ (H \ U (A, ϱ)) = ∂ B1 (0).

(2.6)

The inclusion ϱ A ∈ ∂ B1 (0) is obvious for any point x ∈ ∂ (H \ U (A, ϱ)). Let p ∈ ∂ B1 (0). Then there exists a point a(p) ∈ A such that (p, a(p)) = s(p, A) [17]. Hence x(p) = a(p) + ϱp ∈ ∂ (H \ U (A, ϱ)) and PA (x(p)) = a(p). Thus p = x(p)−ϱ a(p) . By Formulae (2.5), (2.6) and Proposition 1.2 we obtain that the set A is strongly convex of radius R.  x −P x

Theorem 2.2. Let A ⊂ H be a strongly convex subset of radius R > 0. Let x0 , x1 ∈ H \ A, ϱi = ϱ(xi , A), and ai = PA xi , i = 0, 1. Then

∥ a0 − a1 ∥ ≤ 

R



(R + ϱ0 )(R + ϱ1 )

∥x0 − x1 ∥2 − (ϱ0 − ϱ1 )2 .

(2.7)

Proof. Using Proposition 1.2 we get

   x 0 − a0 x 1 − a1    − ∥ a0 − a1 ∥ ≤ R  ϱ0 ϱ1  and

 ∥ a0 − a1 ∥ 2 ≤ R 2 2 − =R

2



2

ϱ0 ϱ1

 (x0 − a0 , x1 − a1 )

∥ a0 − a1 ∥ 2 + ∥ x 0 − x 1 ∥ 2 − ∥ x 1 − a0 ∥ 2 − ∥ x 0 − a1 ∥ 2 2+ ϱ0 ϱ1



.

(2.8)

Let α = ∥a0 − a1 ∥, and ε = ∥x0 − x1 ∥. By Proposition 1.1 we obtain that

 a1 ∈ A ⊂ BR

a0 − R

x 0 − a0



ϱ0

.

Put z = a0 − R 0ϱ 0 . Note that ̸ x0 a0 a1 = π − ̸ za0 a1 , ∥z − a1 ∥ ≤ R. Using the cosine theorem we have 0 x −a

cos ̸ x0 a0 a1 = − cos ̸ za0 a1 = −

R2 + α 2 − ∥z − a1 ∥2 2Rα

≤−

α 2R

and

∥x0 − a1 ∥2 = ∥x0 − a0 ∥2 + ∥a0 − a1 ∥2 − 2∥x0 − a0 ∥ · ∥a0 − a1 ∥ cos ̸ x0 a0 a1 ≥ ϱ02 + α 2 + In a similar way we obtain that

∥x1 − a0 ∥2 ≥ ϱ12 + α 2 +

α 2 ϱ1 R

.

By Formula (2.8) we get

 α ≤R 2

2

α 2 + ε 2 − ϱ12 − α 2 − α Rϱ1 − ϱ02 − α 2 − 2+ ϱ0 ϱ1 2

and after transformations,

α≤ 

R

(R + ϱ0 )(R + ϱ1 )



ε 2 − (ϱ0 − ϱ1 )2 . 

α 2 ϱ0 R

 ,

α 2 ϱ0 R

.

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M.V. Balashov, M.O. Golubev / J. Math. Anal. Appl. 394 (2012) 545–551

Remark 2.1. Note that if x0 ∈ A, i.e. ϱ0 = 0, then Formula (2.7) holds true. In this case a0 = PA x0 = x0 and applying Proposition 1.1 we have

 x0 ∈ A ⊂ BR

a1 − R

Hence cos ̸ x0 a1 x1 ≤ −

x 1 − a1



ϱ1

∥x0 −a1 ∥ 2R

.

as in the proof of Theorem 2.2. By the cosine theorem for the triangle x0 a1 x1 we obtain that

∥x0 − x1 ∥2 = ∥x0 − a1 ∥2 + ϱ12 − 2∥x0 − a1 ∥ϱ1 cos ̸ x0 a1 x1 ≥ ∥x0 − a1 ∥2 + ϱ12 +

ϱ 1 ∥ x 0 − a1 ∥ 2 R

.

The last formula is equivalent to Formula (2.7) with ϱ0 = 0 and x0 = a0 . We see that the estimate (2.7) for an arbitrary strongly convex set of radius R coincides with the estimate (1.2) for the ball of radius R. Corollary 2.1. Let A ⊂ H be a closed convex subset. Then the following properties are equivalent: (1) the set A is strongly convex of radius R > 0, (2) for any ϱ > 0 and any points x0 , x1 ∈ H \ U (A, ϱ), with ai = PA xi , i = 0, 1, the following inequality holds:

∥ a0 − a1 ∥ ≤

R R+ϱ

∥x0 − x1 ∥.

(2.9)

The proof follows by Theorems 2.1 and 2.2. For a bounded subset A ⊂ E put kA = [diam A] + 1. Here [t ] is the largest integer ≤ t. We shall say that a Banach space E satisfies the Mazur intersection property [18–20] if every closed convex bounded subset of the space E coincides with the intersection of all closed balls containing it. Mazur showed that any reflexive Banach space with a Frechet differentiable (on the unit sphere) norm has this property. For a subset A ⊂ E with diam A ≤ R we denote by strcoR A the strongly convex hull of radius R [2,4,11], i.e. strcoR A =



BR (x).

x∈E : A⊂BR (x)

By results [11, Theorem 4.4.2], if diam A ≤ R1 < R2 , then A ⊂ strcoR2 A ⊂ strcoR1 A.

(2.10)

Using Formula (2.10) and the definition of the strongly convex hull, we can write the Mazur intersection property for an arbitrary closed convex bounded subset A ⊂ E as follows: ∞ 

strcok A = A.

k=kA

Let cl w A be the weak closure of a subset A ⊂ E. Every closed convex bounded subset of a reflexive Banach space is weakly compact. This is well-known consequence of the Banach–Alaoglu theorem. Lemma 2.2. Let E be a reflexive Banach space with the Mazur intersection property. Suppose that A ⊂ E is a closed convex bounded subset. Then for any unit vector p ∈ E ∗ we have lim s(p, strcok A) = s(p, A).

k→∞

Proof. Suppose that the statement is false. Without loss of generality we may assume that there exist p0 ∈ ∂ B∗1 (0) and

ε0 > 0 such that

s(p0 , strcok A) − s(p0 , A) ≥ ε0 ,

for all k ≥ kA .

(2.11)

By the reflexivity of the space E we obtain that for each k ≥ kA the set strcok A is weakly compact. Consider a point ak ∈ strcok A with the property (p0 , ak ) = s(p0 , strcok A) and a0 ∈ A with the property (p0 , a0 ) = s(p0 , A). The sequence {ak }∞ k=1 is bounded and without loss of generality, by the Eberlein–Šmulian theorem, we may assume that it weakly converges to some point b0 ∈ E. By the inequality (2.11) we have

(p0 , b0 ) = lim (p0 , ak ) ≥ (p0 , a0 ) + ε0 = s(p0 , A) + ε0 , k→∞

i.e. b0 ̸∈ A.

M.V. Balashov, M.O. Golubev / J. Math. Anal. Appl. 394 (2012) 545–551

549

On the other hand, using the Mazur intersection property and Formula (2.10), we obtain that b0 ∈

∞ 

cl w {ak }∞ k=m ⊂

m=kA

∞ 

strcom A = A. 

m=kA

A Banach space is called strictly convex if the unit sphere in this space contains no nondegenerate line segments. Let ω: [0, +∞) → [0, +∞) be a nonnegative function with the property ω(0) = limt →+0 ω(t ) = 0. We shall say that the metric projection in a Banach space E is uniformly continuous with modulus ω on some class of sets A ⊂ 2E if for any A ∈ A and for any points x0 , x1 ∈ E \ A we have

∥a0 − a1 ∥ ≤ ω(∥x0 − x1 ∥), where ai = PA xi , i = 0, 1. The metric projection in the Hilbert space is uniformly continuous on the class of closed convex sets with modulus

ω(t ) = t.

Theorem 2.3. Let E be a strictly convex reflexive Banach space with the Mazur intersection property. Suppose that the metric projection in the space E is uniformly continuous with modulus ω on the class of strongly convex sets of radius R, for any R ∈ (0, +∞). Then the space E is isomorphic to the Hilbert space. Isomorphism means that there exist another norm ∥ · ∥H in the space E and numbers 0 < c1 ≤ c2 such that (1) c1 ∥x∥H ≤ ∥x∥ ≤ c2 ∥x∥H for all x ∈ E, and (2) (E , ∥ · ∥H ) is the Hilbert space. Proof. Hereafter i means 0 or 1. Let L ⊂ E be an arbitrary closed linear subspace. Choose x0 , x1 ∈ E \ L. The set PL xi is a singleton according to the reflexivity and the strict convexity of the space E. Let ai = PL xi . Put ϱi = ∥xi − ai ∥. Let A = {λa0 + (1 − λ)a1 | λ ∈ [0, 1]}. Note that A ∩ Bϱi (xi ) = {ai } and ai = PA xi . Define a unit vector pi ∈ E ∗ which separates the ball int Bϱi (xi ) and the set A, i.e.

(pi , x) ≥ (pi , ai ) ≥ (pi , a),

for all x ∈ Bϱi (xi ), a ∈ A.

By the definition of the vector pi we have the inclusion

− pi ∈ N (Bϱi (xi ), ai ).

(2.12)

Let Hi = {x ∈ E | (pi , x) = (pi , ai )} and Hi = {x ∈ E | (pi , x) ≥ (pi , ai )}. For each k ≥ kA put B(k) = strcok A, A ⊂ B(k). By Lemma 2.2, +

εki = s(pi , B(k)) − s(pi , A) = s(pi , B(k)) − (pi , ai ) → +0,

k → ∞.

Define εk = max{ε , ε } → +0, k → ∞. We have 0 k

1 k

B(k) ⊂ {x ∈ E | (pi , x) ≤ (pi , ai ) + εk }.

(2.13)

Let xik = PB(k) xi . By the inclusion ai ∈ ∂ Bϱi (xi )∩ A we get xik ∈ Bϱi (xi ) ⊂ Hi+ . Using Formula (2.13) we obtain the following inclusion: B(k) ∩ Hi+ ⊂ Hi + Bεk (0). This implies that xik ∈ Hi + Bεk (0) ∩ Bϱi (xi ).





(2.14)

Without loss of generality, by the Eberlein–Šmulian theorem, we may assume that {xik }∞ k=kA weakly converges to some point bi ∈ E. From the inclusion (2.14) we have

(pi , bi ) = lim (pi , xik ) ≤ (pi , ai ) + lim εk = (pi , ai ) k→∞

k→∞

(2.15)

and also bi ∈ Bϱi (xi ). By the inclusion (2.12) and the strict convexity of the space E we get bi = ai . Thus, {x0k − x1k }∞ k=kA weakly converges to the point a0 − a1 . Uniform continuity of the metric projection with modulus ω for any set B(k) implies the inequality ∥x0k − x1k ∥ ≤ ω(∥x0 − x1 ∥). Using [16, Lemma 27, Section 3, Chapter 2] we have

∥a0 − a1 ∥ ≤ lim inf ∥x0k − x1k ∥ k→∞

and ∥a0 − a1 ∥ ≤ ω(∥x0 − x1 ∥). Thus the metric projection operator onto any closed linear subspace L ⊂ E is uniformly continuous with modulus ω. By results [21, Section 3, Corollary 2] there exists a bounded linear projector from E onto L. Hence every closed linear subspace L ⊂ E is complemented. By the famous theorem of Lindenstrauss (complementary subspace problem [22]) the space E is isomorphic to the Hilbert space. 

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3. Applications Consider the minimization problem of the form min f (x),

x ∈ A ⊂ Rn .

(3.16)

We shall discuss the standard gradient projection algorithm: x1 ∈ ∂ A, xk+1 = PA (xk − αk f ′ (xk )),

α k > 0.

Suppose that: (i) X ⊂ Rn is an arbitrary subset such that A = x∈X BR (x) ̸= ∅. (ii) f : Rn → R is convex, differentiable and the gradient f ′ (x) satisfies the Lipschitz condition with the constant L > 0, i.e. for all x, y ∈ Rn ,



∥f ′ (x) − f ′ (y)∥ ≤ L∥x − y∥. (iii) For any natural k there exists a unit vector p(xk ) ∈ N (A, xk ) such that (p(xk ), f ′ (xk )) ≤ 0 (i.e. xk − αk f ′ (xk ) ̸∈ A for any αk > 0). (iv) The problem (3.16) has a unique solution x∗ ∈ ∂ A. Note that condition (ii) is equivalent to the following one: for all x, y ∈ Rn 1

(x − y, f ′ (x) − f ′ (y)) ≥

L

∥f ′ (x) − f ′ (y)∥2 ;

(3.17)

see [23, Lemma 1.2.3, Theorem 2.1.5]. Theorem 3.1. Suppose that conditions (i)–(iv) hold. Then for αk = α =

∥xk+1 − x∗ ∥ ≤  4

R R2 +

4 L2

∥f ′ (xk )∥2



R + 2L ∥f ′ (x∗ )∥

2 L

we have the estimate

∥xk − x∗ ∥.

(3.18)

Proof. Consider xk ∈ ∂ A. By Proposition 1.1, A ⊂ BR (xk − Rp(xk )), where p(xk ) is from (iii). Let yk = xk − α f ′ (xk ), zk = xk − Rp(xk ). Then

∥yk − zk ∥2 = α 2 ∥f ′ (xk )∥2 + R2 − 2α R(f ′ (xk ), p(xk )) ≥ α 2 ∥f ′ (xk )∥2 + R2 ,  ϱ(yk , A) ≥ ϱ(yk , BR (zk )) = ∥yk − zk ∥ − R ≥ α 2 ∥f ′ (xk )∥2 + R2 − R, ϱ(x∗ − α f ′ (x∗ ), A) = α∥f ′ (x∗ )∥. By Formula (2.7) of Theorem 2.2 and Remark 2.1 (in the case f ′ (x∗ ) = 0) we get

∥xk+1 − x∗ ∥ = ∥PA (xk − α f ′ (xk )) − PA (x∗ − α f ′ (x∗ ))∥ ≤ Ck ∥xk − α f ′ (xk ) − x∗ + α f ′ (x∗ )∥,

(3.19)

where Ck =  4

R R2 +

4 L2

∥f ′ (xk )∥2



R + 2L ∥f ′ (x∗ )∥

.

Using (ii) and (3.17) we obtain that

∥xk − α f ′ (xk ) − x∗ + α f ′ (x∗ )∥2 ≤ ∥xk − x∗ ∥2 − 2α(xk − x∗ , f ′ (xk ) − f ′ (x∗ )) + α 2 ∥f ′ (xk ) − f ′ (x∗ )∥2 2α ′ ∥f (xk ) − f ′ (x∗ )∥2 + α 2 ∥f ′ (xk ) − f ′ (x∗ )∥2 ≤ ∥xk − x∗ ∥2 − L

= ∥xk − x∗ ∥ . 2

Together Formulae (3.19) and (3.20) give (3.18).

(3.20) 

M.V. Balashov, M.O. Golubev / J. Math. Anal. Appl. 394 (2012) 545–551

551

Remark 3.1. If we have some information about vectors p(xk ) (see (iii)) then the estimate (3.18) can be more precise. Using the formula

∥yk − zk ∥2 = α 2 ∥f ′ (xk )∥2 + R2 − 2α R(f ′ (xk ), p(xk )), we obtain that

∥xk+1 − x∗ ∥ ≤  4

R2

R

+

4 L2

∥f ′ (x

k )∥ − 2

4R L

(f ′ (x

 ∥xk − x∗ ∥. 2 ′ (x )∥ ), p ( x )) R + ∥ f k k ∗ L

Remark 3.2. By Nesterov [23, Theorem 2.1.5] we have f (xk ) − f (x∗ ) ≤ (f ′ (x∗ ), xk − x∗ ) +

L 2

L

∥xk − x∗ ∥2 ≤ ∥f ′ (x∗ )∥ · ∥xk − x∗ ∥ + ∥xk − x∗ ∥2 . 2

Remark 3.3. Formula (3.17) is valid for all x, y ∈ A if the condition (ii) holds true for any points x, y ∈ A + (diam A) · B1 (0). This fact easily follows by the proof of Lemma 1.2.3 and Theorem 2.1.5 from [23]. Remark 3.4. The result of Theorem 3.1 holds true in the case of the Hilbert space. Implication (ii) ⇒ (3.17) follows by Polovinkin and Balashov [11, Lemma 1.19.5]. The other steps of the proof for the Hilbert space repeat the proof of Theorem 3.1. Acknowledgments This work was supported by grant RFBR 10-01-00139-a, program ‘‘Development of scientific potential of higher school’’ 2.1.1/11133 and Federal Program ‘‘Kadry’’. We are grateful to a referee for extremely helpful comments and suggestions. References [1] T.J. Abatzoglou, The Lipschits continuity of the metric projection, J. Approx. Theory 26 (1979) 212–218. [2] M.V. Balashov, E.S. Polovinkin, M-strongly convex subsets and their generating sets, Sb. Math. 191 (1) (2000) 27–64. [3] H. Frankowska, Ch. Olech, R-convexity of the integral of the set-valued functions, in: Contributions to Analysis and Geometry, Johns Hopkins Univ. Press, Baltimore, MD, 1981, pp. 117–129. [4] E.S. Polovinkin, Strongly convex analysis, Sb. Math. 187 (2) (1996) 259–286. [5] B.T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. 7 (1966) 72–75. [6] M. Baronti, P.L. Papini, Diameters, centers and diametrically maximal sets, Rend. Circ. Mat. Palermo (2) Suppl. 38 (2) (1995) 11–24. [7] F. Bavaud, Adjoint transform, overconvexity and sets of constant width, Trans. Amer. Math. Soc. 333 (1) (1992) 315–324. [8] H.G. Eggleston, Convexity, Cambridge Univ. Press, Cambridge, 1958. [9] J.P. Moreno, P.L. Papini, R.R. Phelps, New families of convex sets related to diametral maximality, J. Convex Anal. 13 (3–4) (2006) 823–837. [10] E.S. Polovinkin, On the bodies of constant width, Dokl. Math. 397 (3) (2004) 313–315 (in Russian). [11] E.S. Polovinkin, M.V. Balashov, Elements of Convex and Strongly Convex Analysis, Fizmatlit, Moscow, 2007 (in Russian). [12] G.E. Ivanov, E.S. Polovinkin, On strongly convex differential games, Differ. Equ. 31 (10) (1995) 1603–1612. [13] J.-P. Vial, Strong and weak convexity of sets and functions, Math. Oper. Res. 8 (2) (1983) 231–259. [14] M.V. Balashov, G.E. Ivanov, Weakly convex and proximally smooth sets in Banach spaces, Izv. Math. 73 (3) (2009) 455–499. [15] M.V. Balashov, D. Repovš, Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order, J. Math. Anal. Appl. 377 (2) (2011) 754–761. [16] N. Danford, J. Schwartz, Linear Operators, Part I: General Theory, Interscience Publishers, New York, London, 1958. [17] R. James, Characterization of reflexivity, Studia Math. 23 (3) (1964) 205–216. [18] J.R. Giles, D.A. Gregory, B. Sims, Characterization of normed linear spaces with Mazur’s intersection property, Bull. Aust. Math. Soc. 18 (1978) 105–123. [19] A.S. Granero, M. Jiménez-Sevilla, J.P. Moreno, Intersection of closed balls and geometry of Banach spaces, Extracta Math. 19 (1) (2004) 55–92. [20] S. Mazur, Über schwache Konvergenz in den Raümen (Lp ), Studia Math. 4 (1933) 128–133. [21] J. Lindenstrauss, On nonlinear projection in Banach spaces, Michigan Math. J. 11 (3) (1964) 263–287. [22] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces, I, Sequence Spaces, Springer-Verlag, New York, 1977. [23] Yu. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course, Kluwer, 2004.