Paraconvex functions and paraconvex sets

Paraconvex functions and paraconvex sets Huynh Van Ngai∗

and

Jean-Paul Penot†

Abstract We study a class of functions which contains both convex functions and differentiable functions whose derivatives are locally Lipschitzian or H¨olderian. This class is a subclass of the class of approximately convex functions. It enjoys refined properties. We also introduce a class of sets whose associated distance functions are of that type. We discuss the properties of the metric projections on such sets under some assumptions on the geometry of the Banach spaces in which they are embedded. We describe some relations between such sets and functions. Mathematics Subject Classification: 49J52, 46N10, 46T20. Key words: approximately convex function, approximately convex set, monotonicity, nonsmooth analysis, normal, projection, subdifferential

1

Introduction

Some classes of sets or functions are so favorable for smooth and nonsmooth analysis that they have been considered by a number of authors. This is the case for the class of (locally) p-paraconvex functions introduced in [79] and studied in [3], [6], [13], [14], [15], [17], [20], [43], [63], [72], [79]–[92], especially in the case p = 2 in which case the functions are also called semiconvex functions as in [45] and the book [16] which stresses their applications in optimal control and the study of Hamilton–Jacobi equations. This class of functions contains the class of convex functions and the class of functions of class C 2 and enjoys interesting stability properties. In the present paper we study the notion of p-paraconvexity for sets introduced in [92] (in a ∗

Department of Mathematics, Pedagogical University of Quynhon, 170 An Duong Vuong, Qui Nhon, Vietnam † Laboratoire de Math´ematiques Appliqu´ees, CNRS UMR 5142, Facult´e des Sciences, Av. de l’Universit´e 64000 PAU, France

1

global sense and for p = 2), studied in [20, section 5] under the name of property (ω) and called subsmoothness in [6]. We also introduce a notion of intrinsically p-paraconvex set. Both notions are given in terms of the distance function to the set. As in the case of the (more general) classes of approximately convex functions and sets in the sense of [6], [20], [25], [54], [55], [57] which differs from the one used in [35], [34], we provide characterizations. It appears that the classes of p-paraconvex functions and sets form more structured classes in which one has a quantitative control of the variations of the subdifferentials of the functions or of the normal cone to the set at nearby points respectively. For instance, in the class of superreflexive Banach spaces, we show that best approximations to pparaconvex sets exist and we prove the H¨older continuity of projections, obtaining results somewhat similar to the case of convex subsets (see [1], [2], [71] and their references). We also present conditions ensuring that a marginal function is p-paraconvex (see section 4), thus solving an unsettled problem. We relate the concepts of paraconvexity for sets and functions through epigraphs and sublevel sets in section 7. Let us add that for the class of p-paraconvex functions (resp. sets) the usual notions of subdifferential (resp. normal) coincide; this fact make these classes particularly attractive. The concept of p-paraconvexity around a point we study is a localization of a notion introduced in [79]; let us note that the version we study here is not global as in ([15], [79]–[85], [92]). However, it amount to p-paraconvexity in the sense of [79] on some neighborhood of the point. Since it has been proved in [79] that for p > 2 a p-paraconvex function whose restrictions to line segments is absolutely continuous is convex, in the sequel we restrict our attention to the case p ∈ [1, 2]. Definition 1 Given some p ∈ [1, 2], a function f : X → R∞ := R∪{+∞} on a normed vector space X is said to be p-paraconvex around x ∈ dom f := f −1 (R) if there exist c, δ > 0 such that for any x, y ∈ B(x, δ) and any t ∈ [0, 1] one has f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y) + ct(1 − t) kx − ykp .

(1)

The function f is said to be (locally) p-paraconvex on some open subset U of X if it is finite and p-paraconvex around each point of U. The index of p-paraconvexity of f around x is the infimum cf of the set of real numbers c such that there exists some δ > 0 for which relation (1) is satisfied for any x, y ∈ B(x, δ) and any t ∈ [0, 1]. 2

When cf is negative, we have in fact a strong convexity property; the case cf = 0, p = 1 corresponds to approximate convexity (see [54], [55]). The case p = 2 is of special interest, in particular in the setting of Euclidean spaces or Hilbert spaces. As mentioned above, it has been thoroughly studied, at least in finite dimensions or in Hilbert spaces, or in the locally Lipschitzian case. Here we get rid of these restrictions. We relate the notion of p-paraconvexity to the following notion of p-paramonotonicity. For p = 2 it corresponds to a localized version of a concept introduced by A. Pazy [60] (see also [24], [61] for related notions). Definition 2 A multimap (or set-valued map) M : X ⇒ X ∗ is said to be p-paramonotone around x on a subset E of X if there exist some m, δ > 0 such that for any x1 , x2 ∈ E ∩ B(x, δ), x∗1 ∈ M (x1 ), x∗2 ∈ M (x2 ) one has hx∗1 − x∗2 , x1 − x2 i ≥ −m kx1 − x2 kp .

(2)

For E = X one simply says that M is p-paramonotone around x. The index of p-paramonotonicity of M on E around x is the infimum mM of the set of real numbers m > 0 such that there exists some δ > 0 for which relation (2) is satisfied for any x1 , x2 ∈ E ∩ B(x, δ), x∗1 ∈ M (x1 ), x∗2 ∈ M (x2 ). Directional versions of the concepts we study can be introduced, but for the sake of brevity we do not consider them here. We also leave apart the generalization of p-paraconvexity consisting in replacing the term ct(1 − t) kx − ykp in (1) by ct(1 − t)α(kx − yk) where α : R+ → R+ ∪ {+∞} is an hypermodulus, i.e. a nondecreasing function such that α(t)/t → 0 as t → 0+ . The reason for that choice lies in the H¨older estimates we get for the opposite of the distance function to an arbitrary closed subset of a uniformly smooth Banach space (Corollary 13) and for the metric projection onto a p-paraconvex subset of a superreflexive Banach space (Theorem 30); this last topic has been thoroughly studied for convex subsets (see [1], [2] for instance). The notion of p-paraconvex subset is studied in sections 5 and 6. Although the choice p = 2 is the most important one, the case p 6= 2 may have an interest, for instance when working in some Lp space, as illustrated by Proposition 24 below.

3

2

Preliminaries

In the sequel, X is a Banach space with topological dual space X ∗ and S(X) denotes the set of lower semicontinuous (l.s.c.) functions f : X → R∞ := R∪{+∞}. The open ball with center x ∈ X and radius ρ > 0 is denoted by B(x, ρ), while B X (resp. B X ∗ ) stands for the closed unit ball of X (resp. X ∗ ) and SX stands for the unit sphere. Given a subset E of X, the distance function dE associated with E is given by dE (x) := inf e∈E d(x, e). We write E (xn ) → x to mean that (xn ) → x and xn ∈ E for each n ∈ N. Since the functions we consider are not necessarily smooth, we have to use some generalizations of derivatives. A first possibility is to take a primal approach by considering directional derivatives. A second approach is a dual one; it is a one-sided substitute to the derivative called a subdifferential. Several sets of axioms have been devised for such a concept. The notion of subdifferential we adopt here is as versatile as possible: a subdifferential on a Banach space X will be here just a correspondence ∂ : F(X) × X ⇒ X ∗ which assigns a subset ∂f (x) of the dual space X ∗ of X to any f in a subclass F(X) of the set S(X) of lower semicontinuous functions on X and any x ∈ X at which f is finite; we assume it satisfies the following natural property: (M) If f ∈ F(X) is a Lipschitzian function and if x is a minimizer of f , then 0 ∈ ∂f (x). Of course, usual subdifferentials satisfy other conditions which turn them into calculus properties. Among such properties is a form of the mean value theorem we describe below. Another property is the nonemptiness of ∂f (x) for x in a dense subset of X for any Lipschitzian function f on X; then we say that X is a (Lipschitz) ∂-subdifferentiability space. This property is satisfied if X is an Asplund space and ∂ is larger then the Fr´echet subdifferential (sometimes called the firm subdifferential ) defined by x∗ ∈ ∂ F f (x) ⇔ ∀ε > 0 ∃δ > 0, ∀u ∈ δB X : f (x+u) ≥ f (x)+hx∗ , ui−ε kuk . If X is smooth enough, for instance if X is a Hilbert space, and if p ∈]1, 2], this property is satisfied by the p-proximal subdifferential ∂ p ; recall that for p ∈ (1, +∞), the p-proximal subdifferential of a function f on a n.v.s. X at x ∈ X is given by x∗ ∈ ∂ p f (x) ⇔ ∃c > 0, ρ > 0, ∀u ∈ ρB X : f (x+u) ≥ f (x)+hx∗ , ui−c kukp . 4

Clearly ∂ p f (x) is contained in the Fr´echet subdifferential ∂ F f (x). Albeit property (M) is satisfied for ∂ p , for any p > 1, the practical use of ∂ p seems to be limited to the case p ∈]1, 2]. With any subdifferential ∂ is associated a notion of normal cone obtained by setting for a subset E of X and e ∈ E N (E, e) := R+ ∂ιE (e), where ιE is the indicator function of E (given by ιE (x) = 0 if x ∈ E, +∞ otherwise). In the cases ∂ = ∂ D , ∂ = ∂ C and ∂ = ∂ F we get the normal cones N D (E, e), N C (E, e) and N F (E, e) to E at e ∈ E in the senses of Bouligand, Clarke and Fr´echet respectively, as defined below. Other notions have been devised by Ioffe [37], [38], Michel-Penot [51], Mordukhovich [53], Treiman [91] and several other authors. Conversely, to a notion N of normal cone one can associate a notion of subdifferential ∂ by setting Ef := {(x, r) ∈ X × R : r ≥ f (x)}, xf := (x, f (x)) and ∂f (x) := {x∗ ∈ X ∗ : (x∗ , −1) ∈ N (Ef , xf )}. Now let us give the examples of such concepts we will use most. The (contingent or directional or Dini-Hadamard) normal cone N D (E, x) to a subset E of X at x ∈ cl(E) is the polar cone of the directional (or DiniHadamard or Bouligand) tangent cone T D (E, x) to E at x which is the set E of vectors v ∈ X such that there exist sequences (tn ) → 0+ , (xn ) → x for which (tn−1 (xn −x)) → v. Both cones play a crucial role in nonlinear analysis and optimization. The Fr´echet normal cone (or firm normal cone) to E at x is given by x∗ ∈ N F (E, x) ⇔ ∀ε > 0 ∃δ > 0, ∀u ∈ E∩B(x, δ) : hx∗ , u−xi ≤ ε ku − xk . The Clarke normal cone N C (E, x) to a subset E of X at x ∈ cl(E) is the polar cone of the Clarke tangent cone T C (E, x) to E at x which is the set E of vectors v ∈ X such that for any sequence (xn ) → x there exist sequences (tn ) → 0+ and (yn ) in E for which (t−1 n (yn − xn )) → v. The (lower) directional derivative (or contingent derivative or lower DiniHadamard derivative) of f at x is given by f D (x, v) :=

f (x + tw) − f (x) . (t,w)→(0+ ,v) t lim inf

5

The Clarke–Rockafellar derivative or circa-derivative of a function f : X → R = R ∪ {−∞, +∞} finite at x is given by the following formula f (y + tw) − f (y) . r>0 (t,y)→(0 ,x) w∈B(v,r) t +

f C (x, v) := inf lim sup

inf

f (y)→f (x)

Both derivatives are majorized by the dag derivative which is given by f † (x, v) :=

f (y + t(v + x − y)) − f (y) . t (t,y)→(0+ ,x),f (y)→f (x) lim sup

The interest of such a derivative seems to be limited to questions of generalized convexity and to the fact that the associated subdifferential is large. Here, the subdifferential associated with some generalized derivative f 0 is defined by ∂f (x) := {x∗ ∈ X ∗ : x∗ (·) ≤ f 0 (x, ·)}. For f 0 = f D (resp. f 0 = f C ) one recovers the Bouligand or Dini-Hadamard subdifferential ∂ D (resp. Clarke subdifferential ∂ C ). Given a subdifferential ∂, following [44], [53], [65], one can associate to it a corresponding limiting subdifferential ∂ by setting for a l.s.c. function f and a point x of it domain ∂f (x) := w∗ −

lim sup

∂f (u),

(u,f (u))→(x,f (x))

where the w∗ -limsup is the set of weak∗ cluster points of bounded nets (u∗i )i∈I with u∗i ∈ ∂f (ui ), (ui )i∈I → x, (f (ui ))i∈I → f (x). We will make use of mean value inequalities. Under a primal form, such estimates have been pointed out in [62]. In a dual form, as used here, they have been devised by Zagrodny for the Clarke subdifferential [94], [95], and by Loewen [47] for the Fr´echet subdifferential. In [68], such a result is put in a general form including the Ioffe subdifferential [37], [38], the moderate subdifferential [51], and, in Asplund spaces, the Fr´echet subdifferential, the Hadamard subdifferential and the limiting subdifferential. In [5] such a property is obtained under different assumptions. Definition 3 (Fuzzy mean value property) A subdifferential ∂ is said to be valuable on X if for any x ∈ X, y ∈ X\{x}, any lower semicontinuous function f : X → R∪{+∞} finite at x ∈ X and for any r ∈ R such that

6

f (y) ≥ r, there exist u ∈ [x, y) and sequences (un ) → u, (u∗n ) such that u∗n ∈ ∂f (un ), (f (un )) → f (u), lim inf hu∗n , y − xi ≥ r − f (x),

(3)

n

lim inf hu∗n , x − un i ≥ n

kx − uk (r − f (x)) ky − xk

∀x ∈ (x + R+ (y − x)) \[x, u), (4)

lim ku∗n k d(un , [x, y]) n

= 0.

(5)

A simpler version due to Lebourg [46] is as follows. It is satisfied for ∂ := ∂ C , the Clarke subdifferential, ∂ M the moderate subdifferential of [51], or if X is Asplund, by the Bouligand, Fr´echet and limiting subdifferentials. Definition 4 A subdifferential ∂ is said to be Lipschitz-valuable on X if for any x, y ∈ X, any Lipschitzian function f : X → R one can find w ∈ [x, y] and w∗ ∈ ∂f (w) such that f (y) − f (x) = hw∗ , y − xi. We need some results about subdifferentials of distance functions. Lemma 5 ([57], [66]) Let E be a nonempty subset of a Banach space X and let x ∈ E, v ∈ X. Then, one has 1 dE (e + tv). t E , e→x

dC E (x, v) = lim sup t→0+

The second one is close to [11, Lemma 6], [38, Lemma 5], [66, Lemma 1], [58, Lemma 3.6] and [6, Lemma 3.7], but it contains a crucial additional information obtained in [57]. Recall that the norm of X is said to satisfy the Kadec–Klee property if for every x ∈ X, a sequence (xn ) converges to x whenever it weakly converges to x and (kxn k) → kxk . Lemma 6 Suppose that E is a closed nonempty subset of an Asplund space X, w ∈ X\E and w∗ ∈ ∂ F dE (w). Then kw∗ k = 1 and there exist sequences (xn ), (x∗n ) of E and X ∗ respectively such that x∗n ∈ ∂ F dE (xn ) for each n ∈ N and (||xn − wk) → dE (w), (hx∗n , w − xn i) → dE (w),

(||x∗n − w∗ k) → 0.

If moreover X is reflexive and if its norm has the Kadec-Klee property, then a subsequence of (xn ) converges to some best approximation x of w in E and one has hw∗ , w − xi = ||x − wk = dE (w). 7

Now let us consider the limiting normal cone N (E, a) := lim supx→a E N (E, x) F F and the limiting Fr´echet normal cone N (E, a) := lim supx→a E N (E, x) obtained from the normal cone (resp. the Fr´echet normal cone) by the weak upper limit process described above. Corollary 7 ([57]) Let E be a closed nonempty subset of an Asplund space X and let x∗ ∈ ∂ F dE (x) with x ∈ E. Then kx∗ k ≤ 1 and there exist sequences (xn ), (x∗n ) of E and X ∗ respectively such that (xn ) → x, (x∗n ) → x∗ weak∗ and x∗n ∈ ∂ F dE (xn ) for each n ∈ N.

3

Paraconvexity of functions

The following characterization of p-paraconvexity has been given in [19, Thm 5.1] in the case p = 2, X is finite dimensional and the function is Lipschitzian; there, only the proximal subdifferential is used and equivalence with the lower-C2 property in the sense of [77], [88] is proved. This characterization is extended to the case X is a Hilbert space and the function is lower semicontinuous in [6, Thm 4.1], using the Clarke-Rockafellar subdifferential. Here we consider the case of a general Banach space and of an arbitrary valuable subdifferential; the argument for the implication (d)⇒(a) is different. Theorem 8 Let p ∈]1, 2], let f ∈ F(X) and let x ∈ dom f . Suppose the subdifferential ∂f of f is contained in ∂ † f. Then, among the following assertions, one has the implications (a)⇒(b)⇒(c)⇒(d). If moreover ∂ is valuable, all these assertions are equivalent. (a) f is p-paraconvex around x; (b) there exist c ∈ R, ρ > 0 such that for any x ∈ B(x, ρ) ∩ dom f, and any v ∈ B(0, ρ) one has f † (x, v) ≤ f (x + v) − f (x) + c kvkp ; (c) there exist c ∈ R, ρ > 0 such that for any x ∈ B(x, ρ), any x∗ ∈ ∂f (x) and any v ∈ B(0, ρ) one has hx∗ , vi ≤ f (x + v) − f (x) + c kvkp ; (d) ∂f is p-paramonotone around x.

8

(6)

Proof. (a)⇒(b) Given c > cf , let δ > 0 be such that for any y, z ∈ B(x, δ), t ∈]0, 1[ one has f ((1 − t)y + tz) ≤ (1 − t)f (y) + tf (z) + ct(1 − t) ky − zkp . Let us set ρ := δ/2 and let us fix x ∈ B(x, ρ) ∩ dom f and take v ∈ B(0, ρ), y ∈ B(x, ρ) ∩ dom f. Then z := x + v ∈ B(x, δ) and f ((1 − t)y + t(x + v)) − f (y) ≤ f (x + v) − f (y) + c(1 − t) ky − (x + v)kp , t so that, since lim supy→x −f (y) ≤ −f (x), f † (x, v) ≤ f (x + v) − f (x) + c kvkp . (b)⇒(c) is an immediate consequence of the inclusion ∂f ⊂ ∂ † f and of the definition of ∂ † f . (c)⇒(d) is proved in [25, Thm 2] and [6, Thm 4.5] for the Clarke subdifferential; the general case is similar: given c ∈ R, ρ as in (c), for any x, y ∈ B(x, ρ/2), any x∗ ∈ ∂f (x), y ∗ ∈ ∂f (y), taking v := y − x, v 0 := −v we have x, y ∈ dom f and hx∗ , y − xi ≤ f (y) − f (x) + c kx − ykp , hy ∗ , x − yi ≤ f (x) − f (y) + c kx − ykp , hence hx∗ − y ∗ , x − yi ≥ −2c kx − ykp . (d)⇒(a) when ∂ is valuable. Assume ∂f is p-paramonotone around x : there exist some ρ, m > 0 such that for any x, y ∈ B(x, ρ), any x∗ ∈ ∂f (x), y ∗ ∈ ∂f (y) one has hx∗ − y ∗ , x − yi ≥ −m kx − ykp .

(7)

Let δ := ρ/2 and let x, y ∈ B(x, δ), t ∈]0, 1[, z := tx + (1 − t)y, r < f (z). Without loss of generality, to prove (1) we may assume that x 6= y and f (x), f (y) < +∞. Since ∂ is valuable we can find u ∈ [y, z), sequences (un ) → u, (u∗n ) in X ∗ such that u∗n ∈ ∂f (un ) for each n and lim inf hu∗n , n

r − f (y) x − un i> . kx − un k kz − yk

(8)

Let s ∈]0, 1[ be such that z = x + s(u − x) and let zn = x + s(un − x). Since (un ) → u, one has (zn ) → z and since f is l.s.c. at z, for n larger 9

than a certain n0 ∈ N one has un , zn ∈ B(x, ρ) and r < f (zn ). Moreover kzn − xk = (1 − tn ) ky − xk for some sequence (tn ) → t. Using again the fact that ∂ is valuable, we obtain vn ∈ [x, zn ) and sequences (vn,k ) → vn , ∗ ∗ ) such that vn,k ∈ B(x, ρ) and vn,k ∈ ∂f (vn,k ) for any n ≥ n0 , k ∈ N (vn,k and un − vn,k r − f (x) r − f (x) ∗ , i> = . (9) lim inf hvn,k k kun − vn,k k kzn − xk (1 − tn ) ky − xk From relation (8) there exists some n1 ≥ n0 such that for n ≥ n1 one has, since x, zn , un are aligned, hu∗n ,

vn − un x − un r − f (y) r − f (y) i = hu∗n , i> = . kvn − un k kx − un k ky − zk t ky − xk

(10)

On the other hand, since for each n, (vn,k ) → vn , by (10) and (9), for each n ≥ n1 one can find some q(n) such that for k ≥ q(n) one has r − f (y) vn,k − un i> , kvn,k − un k t ky − xk un − vn,k r − f (x) ∗ hvn,k , i> . kun − vn,k k (1 − tn ) ky − xk hu∗n ,

Adding the corresponding sides of these inequalities and using relation (7), we get r − f (y) r − f (x) m kun − vn,k kp−1 ≥ + . t ky − xk (1 − tn ) ky − xk Passing to the limit as k, n → ∞, we get mt(1 − t) ky − xkp ≥ (1 − t)(r − f (y)) + t (r − f (x)) . Since r is arbitrarily close to f (z), we obtain (1).



Corollary 9 Suppose f ∈ F(X) is finite at x ∈ X and p-paraconvex around x with p ∈]1, 2]. Then, for any subdifferential ∂ such that ∂ p f ⊂ ∂f ⊂ ∂ † f and for x near x, one has ∂f (x) = ∂ p f (x) = ∂ † f (x); in particular, ∂ F f (x) = ∂ C f (x). Remark. The proof we have given yields a quantitative approach: the infimum of the constants c appearing in assertion (b) and (c) is majorized by cf and the index of p-paramonotonicity of ∂f around x is majorized by 2cf . On the other hand, the implication (d)⇒(a) shows that cf is majorized by the index of p-paramonotonicity of ∂f around x. We refer to [96] for such estimates in the case of uniformly convex functions which may serve as more precise models.  10

Corollary 10 Suppose f : U → R is a differentiable function on some open subset U of X with a locally (p−1)-H¨ olderian derivative, with p ∈]1, 2]. Then f is locally p-paraconvex on U. Proof. Since f is of class C 1 , its Clarke subdifferential is just the derivative Df of f. Moreover, Df is locally p-paramonotone: for every x ∈ U, there exists some m > 0 and some neighborhood V of x contained in U such that for x, y ∈ V one has hDf (x) − Df (y), x − yi ≥ −m kx − ykp−1 . kx − yk . Thus the result follows from the implication (d)⇒(a) of Theorem 8.

4



Paraconvexity of marginal functions

The statements of the following criteria for p-paraconvexity are simpler than the ones in [55] but they rely on differentiability assumptions. We use the following notation. Given an arbitrary set S, an open subset X0 of a n.v.s. X, x ∈ X0 and g : S × X0 → R, let f : X0 → R be the marginal function f (x) := sup g(s, x)

x ∈ X0 .

s∈S

We assume f is finite at x and for x ∈ X0 , ε > 0, we set S(x, ε) :=

{s ∈ S : g(s, x) ≥ f (x) − ε} {s ∈ S : g(s, x) ≥ ε−1 }

if f (x) < +∞ if f (x) = +∞

T (x, ε) := {s ∈ S : g(s, x) ≥ f (x) − ε}. Theorem 11 If the following conditions are satisfied for some m ≥ 0, p ∈ ]1, 2], ρ > 0, then the marginal function f is Lipschitzian and p-paraconvex on B(x, δ) for some δ ∈]0, ρ[: (C1 ) for each s ∈ S the function gs := g(s, ·) is Fr´echet differentiable on the ball B(x, ρ); (C2 ) for every x ∈ B(x, ρ) one can find some γ := γx > 0 with kgs0 (x)k ≤ m for each s ∈ S(x, γ); (C3 ) there exist some σ, η > 0 such that kgs0 (x) − gs0 (x0 )k ≤ σ kx − x0 kp−1 for any x, x0 ∈ B(x, ρ), s ∈ T (x, η) ∩ T (x0 , η).

11

Proof. Let us apply the characterization of Theorem 8: we want to find some β, σ > 0 such that f C (x, v) ≤ f (x+v)−f (x)+σ kvkp . (11)

∀x ∈ B(x, β), ∀v ∈ βB X

Note that f is l.s.c. on B(x, ρ); we will first show f is Lipschitzian with rate ` := m + ρp−1 σ on some ball B(x, 2δ), where σ, η are as in (C3 ). Let ε ∈]0, 1[ be such that (` + 2)ε < η and let δ ∈]0, ρ/4[∩]0, ε/4[ be such that f (w) > f (x) − ε for each w ∈ B(x, 2δ). Given x, y ∈ B(x, 2δ), we pick γ := γy ∈]0, ε[ such that kgs0 (y)k ≤ m for each s ∈ S(y, γ), as in (C2 ). Given s ∈ S(y, γ) let h : [0, 1] → R and t ∈ [0, 1] be defined by h(t) := g(s, y + t(x − y)), and let t := sup{t ∈ [0, 1] : ∀r ∈ [0, t] h(r) ≥ g(s, y) − `ε}. By continuity of gs , for any t ∈ [0, t], zt := y + t(x − y), we have h(t) = g(s, zt ) ≥ g(s, y) − `ε ≥ f (y) − γ − `ε ≥ f (x) − η, hence s ∈ T (zt , η). In particular, taking t = 0, we see that s ∈ T (y, η) hence, by (C3 ), kgs0 (zt ) − gs0 (y)k ≤ σ kz − ykp−1 . It follows that kgs0 (zt )k ≤ `. The Mean Value Theorem yields some z ∈ [y, y + t(x − y)] such that h(0) − h(t) = tgs0 (z)(x − y).

(12)

Thus h(0) − h(t) ≤ 4t`δ < `ε. If t < 1, by continuity, we have h(t) = g(s, y) − `ε = h(0) − `ε, a contradiction. Therefore t = 1 and (12) yields g(s, y) − g(s, x) = gs0 (z)(x − y) ≤ ` kx − yk .

(13)

Since s is arbitrary in S(y, γ), we get f (y) − f (x) ≤ sup g(s, y) − g(s, x) ≤ ` kx − yk s∈S(y,γ)

and f is Lipschitzian with rate ` on B(x, 2δ). Now let us estimate f C (x, u) for (x, u) ∈ B(x, δ) × SX . Since f is Lipschitzian, we can pick a sequence (xn , tn ) → (x, 0+ ) such that f C (x, u) = 2 limn t−1 n (f (xn + tn u) − f (xn )) . Let us set yn := xn +tn u, γn := min(tn , γyn , ε) and let us choose sn ∈ S(xn + tn u, γn ). Taking n ≥ k, with k large enough, we may suppose that xn ∈ B(x, δ), xn + tn u ∈ B(x, δ). Using relation (13), we obtain that gsn (x) ≥ gsn (yn ) − ` kx − yn k ≥ f (yn ) − γn − `tn , 12

hence, since f is l.s.c. at x, lim inf n gsn (x) ≥ f (x) or (gsn (x)) → f (x). Moreover, using again relation (13), we note that for every w ∈ B(x, δ) and n ≥ k we have gsn (w) ≥ gsn (yn )−` kw − yn k ≥ f (yn )−γn −2`δ ≥ f (x)−2ε−`ε ≥ f (x)−η, (14) so that sn ∈ T (w, η) Taking t ∈]0, δ], from the Mean Value Theorem we get some rn , rn0 ∈ [0, 1] such that 1 (g(sn , x + tu) − g(sn , x)) = gs0 n (x + rn tu)(u), t 1 (g(sn , xn + tn u) − g(sn , xn )) = gs0 n (xn + rn0 tn u)(u). tn Choosing w = x + rn tu and then w = xn + rn0 tn u in relation (14), we get that sn ∈ T (x + rn tu, η) ∩ T (xn + rn0 tn u, η), so that we may apply (C3 ). Thus 1 1 (g(sn , xn + tn u) − g(sn , xn )) − (g(sn , x + tu) − g(sn , x)) tn t 0 0 0 ≤ gsn (xn + rn tn u) − gsn (x + rn tu) p−1

≤ σ kxn − x + rn0 tn u − rn tuk

.

We deduce from these inequalities, the choice of sn and the fact that (g(sn , x)) → f (x) 1 1 (f (xn + tn u) − f (xn )) ≤ lim inf (g(sn , xn + tn u) − g(sn , xn ) + t2n ) n tn n tn   1 p−1 0 ≤ lim inf (g(sn , x + tu) − g(sn , x)) + σ kxn − x + rn tn u − rn tuk n t 1 ≤ (f (x + tu) − f (x)) + σtp−1 kukp−1 . t

lim

Therefore, for any v ∈ δB X , setting v = tu with t ∈]0, δ], u ∈ SX , we get relation (11).  The following immediate consequence partially generalizes [19, Thm 5.1] which is set in finite dimensional spaces and for suprema of functions of class C 2. Corollary 12 Suppose S is a compact topological space, g : S × X → R is continuous with a partial derivative with respect to x ∈ B(x, ρ) which is 13

jointly continuous and Lipschitz with respect to x, uniformly with respect to s ∈ S. Then the marginal function f is Lipschitzian and 2-paraconvex on B(x, δ) for some δ > 0. Another consequence will be used later on. Corollary 13 Suppose X is a uniformly smooth Banach space with modulus of smoothness ρX satisfying for some k > 0, p > 1 the estimate ρX (t) ≤ ktp for t ≥ 0. Then, for any nonempty closed subset E of X the function −dE is p-paraconvex on X\E. Proof. Let j be the derivative of the norm on X\{0}. Let us first prove that, given s > r > 0, there exists some c > 0 such that kj(x) − j(y)k ≤ c kx − ykp−1

∀x, y ∈ sB X \rB X . (15)

By [8, Prop. A.5] we have for any x, y ∈ sB X \rB X

p−1

y p x

kj(x) − j(y)k ≤ k2 − ≤ k2p r2−2p kkyk x − kxk ykp−1 kxk kyk ≤ k2p r2−2p (kyk kx − yk + kyk |kxk − kyk|)p−1 ≤ k22p−1 r2−2p sp−1 kx − ykp−1 . Given x ∈ X\E, let η = ρ = r = dE (x)/2 and let us set S := E, g(e, x) = − kx − ek for x ∈ B(x, ρ), e ∈ E. Then f (x) = −dE (x) for x ∈ B(x, ρ) and for e ∈ T (x, η), i.e. e ∈ E satisfying kx − ek ≤ dE (x) + η we have r ≤ kx − ek ≤ 3r. Thus (15) shows that condition (C3 ) is satisfied. Since the norm is differentiable on B(x, ρ) with derivative j bounded by 1, conditions (C1 ) and (C2 ) are also satisfied. 

5

Paraconvexity of sets

We observe that using the notion of p-paraconvexity for the indicator function ιE of a subset E of X would lead to convexity of E and not to a relaxed form of convexity. Therefore, we rather use the distance function dE . In the sequel x is a point of E and p ∈]1, 2]. Definition 14 A subset E of X is said to be p-paraconvex around x if its associated distance function dE is p-paraconvex around x. 14

This notion is different from the one studied in [50]. Clearly, a pparaconvex set is approximately convex in the sense of [54]. Example. The set E := {(r, s) ∈ R2 : s ≥ |r| − r2 } is p-paraconvex but nonconvex. This example is a special instance of Proposition 25 below. It is not obvious to decide whether the preceding definition depends on the choice of the norm in an equivalent class; on the contrary, the variant presented in the next section will not depend on the choice of the norm inducing the topology. The following result is an easy consequence of [25] when ∂ = ∂ C . However, since we use here an arbitrary subdifferential contained in the Clarke subdifferential, we have to use Theorem 8 with the distance function. Theorem 15 Let ∂ be a subdifferential on the family of Lipschitz functions such that ∂f ⊂ ∂ C f for any Lipschitz function f on X. Then, among the following assertions, one has the implications (a)⇒(b)⇒(c)⇔(c’)⇒(d). If moreover ∂ is Lipschitz-valuable on X all these assertions are equivalent. (a) E is p-paraconvex around x in the sense that dE is p-paraconvex around x; (b) there exist c, ρ > 0 such that for any x ∈ B(x, ρ) and any v ∈ B(0, ρ) one has p dC (16) E (x, v) ≤ dE (x + v) − dE (x) + c kvk ; (c) there exist c, ρ > 0 such that for any x ∈ B(x, ρ), any x∗ ∈ ∂dE (x) and any (u, t) ∈ SX ×]0, ρ) one has hx∗ , ui ≤

dE (x + tu) − dE (x) + ctp−1 ; t

(17)

(c’) there exists c, ρ > 0 such that for any x ∈ B(x, ρ), any x∗ ∈ ∂dE (x) and any v ∈ ρB X one has hx∗ , vi ≤ dE (x + v) − dE (x) + c kvkp ;

(18)

(d) ∂dE is p-paramonotone around x. The following corollary is a consequence of Corollary 9. Corollary 16 If E is p-paraconvex around x for some p > 1, then, for any subdifferential ∂ such that ∂ p f ⊂ ∂f ⊂ ∂ C f for any Lipschitz function f, one has ∂ p dE (x) = ∂ F dE (x) = ∂dE (x) = ∂ C dE (x) for x close to x. 15

6

Intrinsic p-paraconvexity

As in the case of approximate convexity ([57]), the terminology of the definition we adopt now is justified by the fact that the notion we introduce is obtained by restricting the requirement on the distance function to the subset E. Thus, this new notion is more general than the preceding notion. We will show later that the two notions do not coincide. Definition 17 Given p ∈ [1, +∞), a subset E of X is said to be intrinsically p-paraconvex around x ∈ E if there exist c, ρ > 0 such that for any x1 , x2 ∈ E ∩ B(x, ρ), t ∈ [0, 1], one has dE ((1 − t)x1 + tx2 ) ≤ ct(1 − t) kx1 − x2 kp .

(19)

It is intrinsically p-paraconvex if it is intrinsically p-paraconvex around each of its points. Let us note that there is no loss of generality in assuming that E is closed. Let us also note that this definition does not depend on the choice of the norm among the ones inducing the same topology. Characterizations can be given as follows. When one of the assertions (c)-(d) holds, we say that E is ∂-intrinsically p-paraconvex around x. Theorem 18 Let E be a nonempty closed subset of X and let ∂ be a subdifferential such that ∂f ⊂ ∂ C f for any Lipschitz function f on X. Then the following assertions (c), (d) are equivalent and are implied by assertions (a), (b) and (e): (a)⇒(b)⇒(c)⇔(d)⇐(e). When X is a ∂-subdifferentiability space one has (e)⇒(a). (a) E is intrinsically p-paraconvex around x; (b) there exist c, δ > 0 such that for any x, x0 ∈ E ∩ B(x, δ), one has p

0 0 dC E (x, x − x) ≤ c kx − x k ;

(20)

(c) there exist c, δ > 0 such that for any x, x0 ∈ E ∩B(x, δ), x∗ ∈ ∂dE (x), one has p hx∗ , x0 − xi ≤ c kx − x0 k ; (21) (d) ∂dE (·) is p-paramonotone around x on E : there exist c, δ > 0 such that for any x1 , x2 ∈ E ∩ B(x, δ), x∗1 ∈ ∂dE (x1 ), x∗2 ∈ ∂dE (x2 ) one has hx∗1 − x∗2 , x1 − x2 i ≥ −c kx1 − x2 kp ; 16

(22)

(e) there exist c, ρ > 0 such that for any w ∈ B(x, ρ), x ∈ E ∩ B(x, ρ), w ∈ ∂dE (w) one has ∗

dE (w) + hw∗ , x − wi ≤ c kx − wkp .

(23)

Proof. (a)⇒(b) Let ρ > 0 be as in Definition 17 and let x, x0 ∈ E ∩ B(x, ρ). By Lemma 5 , we have 1 dE (e + t(x0 − x)). t E , e→x

0 dC E (x, x − x) ≤ lim sup t→0+

Now, since for t ∈]0, 1[, dE (e + t(x0 − x)) ≤ dE (e + t(x0 − e)) + t ke − xk ≤ ct(1 − t) kx0 − ekp + t ke − xk , we get p

0 0 dC E (x, x − x) ≤ c kx − xk .

(b)⇒(c) is a consequence of the inclusion ∂dE (x) ⊂ ∂ C dE (x) and of the definition of ∂ C dE (x). When ∂ = ∂ C , the reverse implication follows from ∗ ∗ C the relation dC E (x, u) = sup{hx , ui : x ∈ ∂ dE (x)}. (c)⇒(d) Let c, δ > 0 be as in assertion (c) and let x1 , x2 ∈ E ∩ B(x, δ), ∗ x1 ∈ ∂dE (x1 ), x∗2 ∈ ∂dE (x2 ). Taking x = x1 , x∗ = x∗1 , x0 = x2 and adding inequality (21) to the corresponding one obtained by choosing x = x2 , x∗ = x∗2 , x0 = x1 we get relation (22) with c changed into 2c. (d)⇒(c) is obtained in taking x1 = x, x∗1 = x∗ , x2 = x0 , x∗2 = 0 in assertion (d), using the fact that x2 is a minimizer of dE , so that 0 ∈ ∂dE (x2 ). (e)⇒(c) is obvious (take w = x0 ∈ E). (e)⇒(a) when X is a ∂-subdifferentiability space. Given c > 0, ρ > 0 as in assertion (e), let x1 , x2 ∈ E ∩ B(x, ρ), t ∈ [0, 1], w := (1 − t)x1 + tx2 . Since X is a ∂-subdifferentiability space, there exist sequences (wn ) → w, (wn∗ ) such that wn∗ ∈ ∂dE (wn ) for each n ∈ N. Then, as ∂dE (wn ) ⊂ ∂ C dE (wn ) ⊂ B X ∗ , we have (hwn∗ , w − wn i) → 0. Since by convexity w ∈ B(x, ρ), we have wn ∈ B(x, ρ) for n large enough, hence (1 − t)dE (wn ) + (1 − t)hwn∗ , x1 − wn i ≤ (1 − t)c kx1 − wn kp , tdE (wn ) + thwn∗ , x2 − wn i ≤ tc kx2 − wn kp . Adding the corresponding sides of these relations, we get dE (wn ) + hwn∗ , w − wn i ≤ (1 − t)c kx1 − wn kp + tc kx2 − wn kp ,

17

and, passing to the limit, dE (w) ≤ c(1−t)tp kx1 − x2 kp +ct(1−t)p kx2 − x1 kp ≤ 2ct(1−t) kx1 − x2 kp .  Now let us give some specializations to some specific subdifferentials and normal cones. We start with the Fr´echet normal cone. Corollary 19 For the Fr´echet subdifferential ∂ F , assertion (c) of the preceding theorem is equivalent to the following assertion: (f ) there exists c, δ > 0 such that for any x, x0 ∈ E ∩ B(x, δ), x∗ ∈ N F (E, x) one has p

hx∗ , x0 − xi ≤ c kx∗ k kx − x0 k .

(24)

When (c) holds with ∂ = ∂ C , for all x ∈ E ∩ B(x, δ) one has N C (E, x) = N F (E, x). If X is an Asplund space then all the assertions of Theorem 18 are equivalent to (f ). Proof. The relation ∂ F dE (x) = N F (E, x) ∩ B X ∗ for each x ∈ E, yields equivalence of assertions (c) and (f ) by an homogeneity argument which does not require any assumption on X. Moreover, assertion (c) shows that ∂ C dE (x) = ∂ F dE (x) for each x ∈ E ∩ B(x, δ) and that ∂ F dE (x) is weak* closed. Therefore, by [18, Prop. 2.4.2], denoting by co∗ (S) the weak∗ closed convex hull of a subset of X ∗ , we get N C (E, x) = co∗ (R+ ∂ C dE (x)) = co∗ (R+ ∂ F dE (x)) = N F (E, x) since N F (E, x) is convex and weak∗ closed by (f). (c)⇒(e) When X is an Asplund space. Let c, δ > 0 be as in (c). Let ρ := δ/2 and let w ∈ B(x, ρ)\E (note that (23) coincides with (21) if w ∈ E), x ∈ E ∩B(x, ρ), w∗ ∈ ∂ F dE (w). By Lemma 6 we can find sequences (xn ) in E and (x∗n ) in X ∗ such that x∗n ∈ ∂ F dE (xn ) for each n ∈ N and (||x∗n − w∗ k) → 0. (25) Since dE (w) ≤ ||x − wk < ρ and ||x − wk > 0, we may suppose that ||xn − wk ≤ ρ, ||xn − wk ≤ 2||x − wk for each n ∈ N; then xn ∈ B(x, 2ρ) ⊂ B(x, δ) and ||x − xn k ≤ ||x − wk + kw − xn k ≤ 3||x − wk. Now, for n large (||xn − wk) → dE (w),

(hx∗n , w − xn i) → dE (w),

18

enough, relation (25) implies the inequality of the first line below, while assertion (c) ensures the passage from the second line to the third one: dE (w) + hw∗ , x − wi ≤ lim sup (hx∗n , w − xn i + (hx∗n , x − wi + ||x∗n − w∗ k kx − wk)) n

≤ lim sup (hx∗n , x − xn i + ||x∗n − w∗ k kx − wk) n

≤ lim sup (c kx − xn kp + c kx − wkp + ||x∗n − w∗ k kx − wk) . n

Passing to the limit, and using the estimate ||x − xn k ≤ 3||x − wk obtained above, we get dE (w) + hw∗ , x − wi ≤ c(3p + 1) kx − wkp . Finally, since X is Asplund, it is a ∂ F -subdifferentiability space, so that (e) ensures that E is intrinsically p-paraconvex by Theorem 18.  The case of the limiting subdifferential can easily be derived from the preceding corollary. Corollary 20 Suppose that E is a closed subset of an Asplund space X and let ∂ be the limiting Fr´echet subdifferential ∂ F . Then all the assertions of Theorem 18 are equivalent. Proof. Using Corollary 7, the passages from the assertions using ∂ F to the assertions involving ∂ F result from a passage to the weak∗ limit for a bounded net; the reverse implications are obvious.  Now let us turn to the Clarke subdifferential. It would be interesting to know whether one can get rid of the assumption of the last assertion of the next corollary that X is an Asplund space. Corollary 21 For a closed subset E of a Banach space X and ∂ = ∂ C , among the assertions of Theorem 18 the following implications hold: (e)⇒(a)⇒(b)⇔(c)⇔(d). If moreover X is an Asplund space, all these assertions are equivalent. Proof. The implications follow from the choice ∂ = ∂ C in Theorem 18 since any Banach space is a ∂ C -subdifferentiability space. When X is an Asplund space the equivalences can be deduced from the preceding corollary: since ∂ C dE (x) = co∗ (∂ F dE (x)) one has the equivalence (eF )⇔(eF )⇔(eC ) (where (eC ), (eF ), (eF ) are (e) for ∂ = ∂ C , ∂ F , ∂ F respectively), hence (cC )⇒(cF )⇒(eF )⇒(eC )⇒(a).  19

The preceding local property can be related to a global one as defined and studied in [15], [20], [29], [28], [31] in the case p = 2, X being a Hilbert space. Definition 22 Given a subset E of X, p ∈]1, 2] and a continuous function ϕ : E × E → R+ , the subset E of X is said to be ϕ − p-convex if for any x, y ∈ E and x∗ ∈ N F (E, x) one has hx∗ , y − xi ≤ ϕ(x, y) kx∗ k kx − ykp .

(26)

In this definition the choice of the Fr´echet normal cone is natural since if this definition holds for some other cone N (E, x) contained in N C (E, x) one has N (E, x) ⊂ N F (E, x). The following result clarifies the links between ϕ − p-convexity and p-paraconvexity. Proposition 23 Let p ∈]1, 2], a subset E of a Banach space X, a continuous function ϕ : E × E → R+ such that E is ϕ − p-convex. Then E is ∂ F -intrinsically p-paraconvex around each point of E in the sense that assertion (c) of Theorem 18 is satisfied with ∂ = ∂ F . Conversely, if E is ∂ F -intrinsically p-paraconvex around each point of E, then E is ϕ − p-convex for some continuous function ϕ : E × E → R+ . Proof. The first assertion is immediate: given c > ϕ(x, x) for some x ∈ E, relation (21) is satisfied whenever δ > 0 is small enough. Let us prove the converse by first showing that there exist an open neighborhood V of the diagonal of E × E and a continuous function ψ : W → R+ such that for any (x, y) ∈ V and x∗ ∈ N F (E, x) one has hx∗ , y − xi ≤ ψ(x, y) kx∗ k . kx − ykp .

(27)

Given (x, y) ∈ E × E, let ωp (x, y) = sup{hx∗ ,

y−x i : x∗ ∈ N F (E, x) ∩ B X ∗ }. kx − ykp

Assertion (c) of Theorem 18 shows that for any e ∈ E there exist c > 0, ρ > 0 such that ωp (x, y) ≤ c for any x, y ∈ B(e, ρ). Clearly, the infimum cp (e) of such constants c defines an upper semicontinuous function on E. Since E is a metric space, there exists a continuous function ψp : E → R such that cp (e) + 1 ≤ ψp (e) for every e ∈ E. Then, by definition of cp , for 20

each e ∈ E one can find some ρ(e) > 0 such that ωp (x, y) ≤ cp (e)+1 ≤ ψp (e) for every x, y ∈ B(e, ρ(e)). Since E is paracompact, we can find a locally finite open covering (Ve )e∈E of E such that Ve ⊂ B(e, ρ(e)) for every e ∈ E. Let us set [ X V := Ve × Ve , ψ(x, y) := λe (x, y)ψp (e), e∈E

e∈E

where (λe )e∈E is a continuous partition of unity subordinated to the covering (Ve × Ve )e∈E of V. Then ψ is continuous and for every (x, y) ∈ V one has X X ωp (x, y) = λe (x, y)ωp (x, y) ≤ λe (x, y)ψp (e) = ψ(x, y), e∈E

e∈E

so that inequality (27) holds. Let U be a closed neighborhood of the diagonal of E × E contained in V and let ϕ be given by ϕ(x, y) = (1 − t(x, y))ψ(x, y) + t(x, y) where t(x, y) :=

1 kx − ykp−1

d((x, y), U ) . d((x, y), (E × E) \V ) + d((x, y), U )

Then ϕ is a continuous function on E ×E. For (x, y) ∈ U, x∗ ∈ N (E, x), one y−x ∗ ∗ ∗ has hx∗ , kx−yk p i ≤ kx k ψ(x, y) = kx k ϕ(x, y). For (x, y) ∈ (E × E) \V, x ∈ y−x 1 ∗ = kx∗ k ϕ(x, y). For (x, y) ∈ N (E, x), one has hx∗ , kx−yk p i ≤ kx k kx−ykp−1 y−x 1 ∗ , kx∗ k ψ(x, y)) ≤ V \U, x∗ ∈ N (E, x), one has hx∗ , kx−yk p i ≤ min(kx k kx−ykp−1 kx∗ k ϕ(x, y). Therefore relation (26) is satisfied.  The following statement can be considered as an illustration of the use of a global property as in the preceding definition.

Proposition 24 Let p ≥ q ∈ [1, ∞), let (S, S, µ) be a measured space with finite measure, let c ∈ L∞ (S, R) and let F : S ⇒ Rn be a measurable multifunction such that ∀s ∈ S, ∀x, y ∈ F (s), ∀t ∈ [0, 1]

dF (s) ((1−t)x+ty) ≤ c(s)t(1−t) kx − ykp/q .

Then the set E of measurable selections of F (·) which are in Lp (S, Rn ) considered as a subset of X := Lq (S, Rn ) is such that ∀u, v ∈ E, ∀t ∈ [0, 1]

dX ((1−t)u+tv, E) ≤ kck∞ t(1−t) ku − vkp/q p . (28) 21

Proof. Let u, v ∈ E, and for t ∈ [0, 1], let w := (1 − t)u + tv. Then, by [78, Thm 14.60], we can interchange minimization and integration to get Z Z q q dX (w, E) := inf kw(s) − z(s)k µ(ds) = inf kw(s) − ekq µ(ds) z∈E S e∈F (s) S Z ≤ c(s)q tq (1 − t)q ku(s) − v(s)kp µ(ds) S

≤ kckq∞ tq (1 − t)q ku − vkpp . Inequality (28) follows.

7

Links between p-paraconvex sets and functions

Some links between geometrical properties and analytical properties are contained in the next statements. In the present section we suppose p > 1. Unless otherwise specified, we endow the product space X := W × R of a n.v.s. W with R with a product norm, i.e. a norm such that the projections and the insertions w 7→ (w, 0) and r 7→ (0, r) are nonexpansive. Then, for each (w, r) ∈ W × R we have max (kwk , |r|) ≤ k(w, r)k ≤ kwk + |r| . Proposition 25 Let W be a normed vector space and let f : W → R∞ be a l.s.c. function which is p-paraconvex around w ∈ W. Then, for any r ≥ f (w) the epigraph E of f is intrinsically p-paraconvex around x := (w, r). Proof. Let c, ρ > 0 be such that f ((1 − t)w1 + tw2 ) ≤ (1 − t)f (w1 ) + tf (w2 ) + ct(1 − t) kw1 − w2 kp for any w1 , w2 ∈ B(w, ρ), t ∈ [0, 1]. Let xi := (wi , ri ) (i = 1, 2) be elements of the epigraph E of f in B(x, ρ) and let t ∈ [0, 1], w := (1 − t)w1 + tw2 , r := (1−t)r1 +tr2 , x := (w, r). Then, as w1 , w2 ∈ B(w, ρ) and (w, f (w)) ∈ E, one has dE (x) = 0 if f (w) ≤ r and dE (x) ≤ k(w, r) − (w, f (w))k ≤ f (w) − r if f (w) > r, so that dE (x) ≤ max(0, f (w) − r) ≤ max(0, (1 − t)(f (w1 ) − r1 ) + t(f (w2 ) − r2 ) + ct(1 − t) kw1 − w2 kp ) ≤ ct(1 − t) kw1 − w2 kp ≤ ct(1 − t) kx1 − x2 kp . Thus E is intrinsically p-paraconvex around x. Let us complete the preceding result with the following one. 22



Proposition 26 Let f : W → R be a function which is Lipschitzian with rate ` > 0 on some ball B(w, ρ). Suppose X := W × R is endowed with the norm given by k(w, r)k = ` kwk + |r| . If f is p-paraconvex around w, then, for any r ≥ f (w), the epigraph E of f is p-paraconvex around x := (w, r). Proof. We may suppose r = f (w) since otherwise x is in the interior of E. By [33], [37] one can find ρ0 ∈]0, ρ[ such that dE (w, r) = (f (w) − r)+ for (w, r) ∈ B(x, ρ0 ) with x := (w, f (w)), X being endowed with the norm described in the statement; here, for t ∈ R, t+ stands for max(t, 0). Let c > 0 and δ ∈]0, ρ0 [ be such that f ((1 − t)w1 + tw2 ) ≤ (1 − t)f (w1 ) + tf (w2 ) + ct(1 − t) kw1 − w2 kp for any w1 , w2 ∈ B(w, δ), t ∈ [0, 1]. Let xi := (wi , ri ) ∈ B(x, δ) for i = 1, 2 and let w := (1 − t)w1 + tw2 , r := (1 − t)r1 + tr2 , x := (w, r). Then we have w1 , w2 ∈ B(w, δ) and f (w) − r ≤ (1 − t)(f (w1 ) − r1 )+ + t(f (w2 ) − r2 )+ + ct(1 − t) kw1 − w2 kp hence dE (x) ≤ (1 − t)dE (x1 ) + tdE (x2 ) + c`−p t(1 − t) kx1 − x2 kp .  Let us give a kind of converse to the preceding propositions. Theorem 27 Let W be a Banach space and let f : W → R be a function which is locally Lipschitzian around w ∈ W and such that the epigraph E of f is an intrinsically p-paraconvex subset of X := W × R around x := (w, f (w)). Then f is a p-paraconvex function around w. Proof in the case W is an Asplund space. In view of the characterization of p-paraconvexity of a function given in Theorem 8, it suffices to prove that ∂ F f is p-paramonotone around w. Let ` be the Lipschitz rate of f on some ball B(x, ρ0 ). By Corollary 19 there exist c > 0 and some ρ ∈]0, ρ0 [ such that for any x1 , x2 ∈ E ∩ B(x, ρ) and any x∗1 ∈ N F (E, x1 ), x∗2 ∈ N F (E, x2 ) one has hx∗1 − x∗2 , x1 − x2 i ≥ −c kx1 − x2 kp . 23

Let ρ0 := ρ/(` + 1). Then, for w1 , w2 ∈ B(w, ρ0 ), wi∗ ∈ ∂ F f (wi ) for i = 1, 2, setting xi := (wi , f (wi )), x∗i := (wi∗ , −1), one has xi ∈ B(x, ρ) and x∗i ∈ N F (E, xi ), hence hw1∗ −w2∗ , w1 −w2 i = hx∗1 −x∗2 , x1 −x2 i ≥ −c kx1 − x2 kp ≥ −c(`+1)p kw1 − w2 kp . Thus f is p-paraconvex around w.  The proof in the general case relies on the following lemma extracted from the proof of [66, Prop. 10 c]; it is close to previous results of that kind due to F.H. Clarke [18] and to A.D. Ioffe [38] in the case x = (w, f (w)). Lemma 28 Let f : W → R be a function which is Lipschitzian with rate ` on a ball B(w, ρ) of W. Then, for σ ∈]0, ρ[ small enough and for any w ∈ B(w, σ) and any w∗ ∈ ∂ C f (w) one has (w∗ , −1) ∈ ∂ C dE (x), where E is the epigraph of f and x := (w, f (w)), X := W × R being endowed with the norm given by k(w, r)k = ` kwk + |r| . Proof of the theorem in the general case. Since intrinsic p-paraconvexity is preserved when using an equivalent norm, we may use the norm described in the lemma and take ρ, σ > 0 as there. We use the implication (a)⇒(c) of Corollary 21: there exist c > 0, γ ∈]0, σ[ such that for any x, x0 ∈ E ∩ B(x, γ), x∗ ∈ ∂ C dE (x), one has p

hx∗ , x0 − xi ≤ c kx − x0 k .

(29)

Let δ := γ/4`. Now, by the preceding lemma, for w ∈ B(w, δ), u ∈ δB W , w0 := w + u ∈ B(w, 2δ) and any w∗ ∈ ∂ C f (w) we have x := (w, f (w)) ∈ E ∩ B(x, γ), x0 := (w0 , f (w0 )) ∈ E ∩ B(x, σ), and (w∗ , −1) ∈ ∂ C dE (x). Then, by inequality (29), p

hw∗ , ui − (f (w0 ) − f (w)) ≤ c (` kw0 − wk + |f (w0 ) − f (w)|) ≤ c2p `p kukp . Thus, for any ε > 0, there exists δ > 0 such that for any w ∈ B(w, δ), any w∗ ∈ ∂ C f (w) and any u ∈ B(0, δ) one has f (w + u) − f (w) ≥ hw∗ , ui −  c2p `p kukp , so that f is p-paraconvex around w by Theorem 8. Finally, let us turn to sublevel sets. Proposition 29 Let X be an Asplund space and let f : X → R∞ . Suppose f is continuous and p-paraconvex around x ∈ S := {x ∈ X : f (x) ≤ 0} and there exist b > 0, r > 0 such that kx∗ k ≥ b for each x ∈ (X\S) ∩ B(x, r) and each x∗ ∈ ∂ F f (x). Then S is intrinsically p-paraconvex around x. 24

Proof. Without loss of generality we may suppose f takes the value +∞ on X\U, where U := B(x, r). Then, by [97], [21], [41], [64, Thm 9.1], [58, Thm 3.2] and several other contributions, we have f+ (x) ≥ bdS (x) for x ∈ U for f+ := max(f, 0). Let ε > 0 be given. Using Theorem 8, we can find c > 0, δ ∈]0, r[ such that hx∗ , x0 − xi ≤ f (x0 ) − f (x) + c kx0 − xk

p

for any x, x0 ∈ B(x, δ), x∗ ∈ ∂ F f (x). Given x, x0 ∈ S∩B(x, δ), x∗ ∈ ∂ F dS (x), using [58, Cor. 4.1] we can find sequences (λn ), (xn ), (x∗n ) in [0, 1], X, X ∗ respectively such that (xn ) → x, (λn x∗n ) → x∗ and x∗n ∈ ∂ F f (xn ) for each n ∈ N. Let us first suppose x∗ 6= 0. Then f (x) = 0 because f is continuous at x and x cannot belong to the interior of S. For n so large that xn ∈ B(x, δ), we have p hx∗n , x0 − xn i ≤ f (x0 ) − f (xn ) + c kx0 − xn k . Thus p


hx∗ , x0 −xi = limhλn x∗n , x0 −xn i ≤ lim sup λn f (x0 ) − f (xn ) + c kx0 − xn k n

n

since f (x0 ) ≤ 0, λn ∈ [0, 1] and (f (xn )) → 0. When x∗ = 0, the inequality hx∗ , x0 − xi ≤ c kx0 − xkp is obvious. Thus, assertion (c) of Corollary 19 is  satisfied for E := S and S is intrinsically p-paraconvex around x.

8

A counter-example

Because the concepts of paraconvexity and intrinsic paraconvexity seem to be so close, one may wonder whether they coincide. The following counterexample shows that this is not the case. Given p = 3/2, let E be the hypograph of the function f : r 7→ |r|p from R to R : E := {(r, s) ∈ R2 : s ≤ |r|p }. Let us endow R2 with the Euclidean norm and show that E is intrinsically paraconvex around (0, 0), but not paraconvex around (0, 0). Since E is the image by the isometry (r, s) 7→ (r, −s) of the epigraph of the function −f whose derivative is (p − 1)-H¨olderian, it is intrinsically p-paraconvex around (0, 0). Let (r, s) ∈ R2 be close to (0, 0), with r ≥ 0, s > rp . In order to

25

p

≤ c kx0 − xk

compute d2E (r, s) we have to minimize the function x 7→ (x − r)2 + (xp − s)2 . The necessary condition is x − r + pxp−1 (xp − s) = 0.

(30)

Let y := x1/2 . The preceding equation is equivalent to 2y 2 − 2r + 3y(y 3 − s) = 0.

(31)

Fixing s, this equation defines y := x1/2 as an implicit function of r; moreover, one has y(0) = y0 , where y0 > 0 is the unique solution of the equation 3y 3 + 2y − 3s = 0. The implicit function theorem asserts that y 0 (0) is determined by the relation   y 0 (0) 4y0 + 12y03 − 3s = 2 obtained by deriving in (31). Thus, since x0 (0) = 2y 0 (0)y0 and 3s = 3y03 +2y0 by (31), 4 . x0 (0) = 2 + 9x(0) Therefore, for some function r 7→ q(r) with limit x0 (0) as r → 0+ , we have d2E (r, s) = (x0 + q(r)r − r)2 + ((x0 + q(r)r)3/2 − s)2 = x20 + 2x0 (q(r) − 1)r + (q(r) − 1)2 r2 + (x0 + q(r)r)3 − 2s(x0 + q(r)r)3/2 + s2 1/2

3/2

= (x20 + x30 − 2sx0 + s2 ) + 2x0 (q(r) − 1)r + 3x20 q(r)r − 3sx0 q(r)r + o(r) 1/2

and since, by (30), 3sx0 = 3x20 + 2x0 , taking limits as r → 0, we get
1/2 r−1 d2E (r, s) − d2E (0, s) → 2x0 (x0 (0) − 1) + 3x20 x0 (0) − 3sx0 x0 (0) = −2x0 . Thus, given a ∈]0, 2x0 [, for r > 0 small enough, we have d2E (r, s) − d2E (0, s) ≤ −ar.

(32)

On the other hand, if dE is p-paraconvex, one can find c, δ > 0 such that for k(r, s)k ≤ δ one has 1 1 1 dE (0, s) ≤ d(r, s) + d(−r, s) + c(2r)p . 2 2 4 Now, since E is invariant by the symmetry (r, s) 7→ (−r, s), dE (r, s) ≥ dE (0, s) − 2p−2 crp and for r sufficiently small the right hand side of this inequality is positive, so that
2 d2E (r, s) ≥ dE (0, s) − 2p−2 crp , d2E (r, s) − d2E (0, s) ≥ −2p−1 crp dE (0, s) + 22p−4 c2 r2p , a contradiction with (32) when k(r, s)k is sufficiently small. 26

9

Paraconvex sets and projections

The following result shows that, in the context of super-reflexive spaces, p-paraconvexity of a distance function is equivalent to its continuous differentiability with a H¨olderian derivative. It is reminiscent to [19, Thm 4.8] which takes place in a Hilbert space. There, p = 2 and a Lipschitz property is given instead of a H¨olderian property. Moreover, here U is not a uniform enlargement of E; it may be small (or large) and far from E. Theorem 30 Let X be a super-reflexive Banach space, let E be a closed subset of X and let U be an open subset of X. The following assertions relative to some choice of p ∈]1, 2] and of an equivalent norm on X are equivalent: (a) Each w ∈ U has a unique metric projection PE (w) in E and the mapping w 7→ PE (w) is locally H¨olderian on U \ E. (b) dE (·) is differentiable with a locally H¨ olderian derivative on U \ E. (c) For each w ∈ U \ E, dE (·) is p-paraconvex around w for some p ∈ ]1, 2]. Proof. (a)⇒(b) As in [57, Thm 21], for any w ∈ U \E, we can show that dE is differentiable and d0E (w) = j(w − PE (w)), where j := (k·k)0 . Let us prove that d0E (.) is locally (p − 1)-H¨olderian on U \ E, taking on X an equivalent norm of type p, i.e. such that its modulus of smoothness ρX satisfies (for some k > 0) ρX (t) ≤ ktp for t ∈ R+ , what is possible by [8, Thm A.6]. Then by relation (15), for any s > r > 0 there exists some c > 0 such that kj(x) − j(y)k ≤ c kx − ykp−1

∀x, y ∈ sB X \rB X .

Let w ∈ U \E be given. By assumption, there exist γ ∈]0, 1], ` > 0, δ > 0 such that kPE (u) − PE (w)k ≤ ` ku − wkγ

∀u, w ∈ B(w, δ).

By the above relations, using the fact that for w close enough to w the number kw − PE (w)k lies in some interval [r, s] with s > r > 0, there exist 27

c, ρ > 0 such that for all u, w ∈ B(w, ρ), one has kd0E (u) − d0E (w)k = kj(u − PE (u)) − j(w − PE (w))k ≤ c k(u − PE (u)) − (w − PE (w))kp−1 ≤ c(ku − wk + kPE (u) − PE (w)k)p−1 ≤ c(ku − wk + ` ku − wkγ )p−1 ≤ c(2ρ1−γ + `)p−1 ku − wkγ(p−1) . (b)⇒(c) is a consequence of Corollary 10. (c)⇒(b) Since X is super-reflexive, it can be endowed with an equivalent uniformly smooth norm with a modulus of smoothness ρX satisfying for some k > 0, p > 1 the estimate ρX (t) ≤ ktp for t ≥ 0. By Corollary 13, the function −dE is p-paraconvex. Thus, with the assumption of (c), ∂ F (−dE )(w) 6= ∅ and ∂ F dE (w) 6= ∅ at any w ∈ U \ E. Therefore, dE (·) is Fr´echet differentiable on U \ E. Let us prove that d0E (·) is locally H¨olderian on U \ E. Let x ∈ U \E and assume that dE (·) is p−paraconvex around x. By the p−paraconvexity of dE (·), −dE (·), by virtue of Theorem 8, there exist c, δ > 0 such that for all u, w ∈ B(x, δ), v ∈ δB X , one has −hd0E (u), vi 6 −dE (u + v) + dE (u) + c||v||p , hd0E (w), vi 6 dE (w + v) − dE (w) + c||v||p . Thus, for any u, w ∈ B(x, δ), for all v ∈ B(0, δ), we have hd0E (w) − d0E (u), vi 6 2||u − w|| + 2c||v||p . Hence ||d0E (u) − d0E (w)|| 6 1

2||u − w|| + 2ctp−1 , t 1

1

for all t ∈]0, δ[. Taking t = 2− p δ 1− p ku − wk p ∈]0, δ[, we obtain ||d0E (u)−d0E (w)||

 6

1+ p1

2

δ

1−p p

1 p

+2 δ

(p−1)2 p

 c ||u−w||

p−1 p

∀u, w ∈ B(x, δ).

(b)⇒(a) By a result of Pisier (see [8, Thm A.6], [76], [74], [8, Thm A.6]) there exists some equivalent norm on X and c > 0, q ≥ 2 such that 1 δX (t) := inf{1 − kx + yk : x, y ∈ B X , kx − yk ≥ t} ≥ ctq ∀t ∈ [0, 2]. 2 Moreover, by [30, Prop. 5.2 p.159] we may assume k·k is differentiable on X\{0}. Now, by [93, Thm 1], given s > r > 0, one can find some b > 0 such that for any x, y ∈ sB X \rB X one has hJ(x) − J(y), x − yi ≥ bδX (kx − yk /2s), 28

where J(·) :=

1 2

k·k2


0

= k·k j(·). It follows that for such x, y one has

kJ(x) − J(y)k ≥ 2−q s−q bc kx − ykq−1

(33)

By [57], the metric projection mapping PE (·) is single-valued and continuous on U \ E. Furthermore, d0E (w) = j(w − PE (w)) for all w ∈ U \ E,
0 hence 21 d2E (w) = J(w − PE (w)). Let us prove that w 7→ PE (w) is locally
0 H¨olderian on U \ E. Since d0E is locally H¨olderian, 21 d2E is also locally H¨olderian as




2
0

2 0

dE (x) − dE (y) ≤ 2dE (x) kd0E (x) − d0E (y)k + 2 kd0E (y)k kx − yk . Given x ∈ U \E, assumption (b) yields some γ ∈]0, 1], a, δ > 0 such that kJ(u − PE (u)) − J(w − PE (w))k ≤ a ku − wkγ

∀u, w ∈ B(x, δ).

Then inequality (33) yields k(u − PE (u)) − (w − PE (w))kq−1 ≤ 2q sq b−1 c−1 a ku − wkγ , hence kPE (u) − PE (w)k ≤ ku − wk + 2q/q−1 sq/q−1 b1/1−q c1/1−q a1/q−1 ku − wkγ/q−1 . This shows that the metric projection PE (·) is locally H¨olderian around x.  Acknowledgements. The authors are grateful to C. Z˘alinescu for a careful reading of the manuscript and to an anonymous referee who pointed out some additional references and limitations. The first author would like to thank the Department of Mathematics of the University of Pau for hospitality and support.

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