Journal of Convex Analysis Volume 1 (1994), No.1, 87–100

Epiconvergence and ε-Subgradients of Convex Functions

Andrei Verona

Department of Mathematics, California State University Los Angeles, CA 90032, e-mail: averona@calstatela.edu

Maria Elena Verona

Department of Mathematics, Pasadena City College 1570 E. Colorado Blvd., Pasadena, CA 91106 e-mail: everona@alumni.caltech.edu

Received 1 December 1993 Revised manuscript received 10 May 1994

We study, in the context of normed spaces (not necessarily Banach), equivalences between the epiconvergence of convex functions and the set-convergence of their ε-subdifferentials.

Keywords : convex function, subdifferential, ε-subdifferential, slice convergence, Attouch-Wets convergence, Kuratowski-Painlevé convergence.

1991 Mathematics Subject Classification: 46N10, 26B25, 40A30, 52A41, 54B20.

1. Introduction

Recently several new topologies on the set of convex subsets of a normed space were introduced and studied, the two most important ones being the Attouch-Wets topology (see e.g. [4], [5], [6]) and the slice topology (see e.g. [2], [7], [8]). What is notable about both of them is that most of their properties (e.g. continuity of the polarity and conjugacy maps) not only hold in Banach spaces, but also in general normed spaces. However, there are other notions, (e.g. the subdifferential map) which are not useful outside the Banach context but were used to characterize these topologies. For example (in the Banach case), convergence in the Attouch-Wets topology was characterized in [10] in terms of the behaviour of a certain operator involving the subdifferential map, while in [2] convergence in the slice topology was characterized in terms of the Kuratowski-Painlevé convergence of the subdifferential map. Thus it seems appropriate to reformulate these characterizations in the context of normed spaces, not necessarily Banach. This is what we do in this paper: we show that the results of Attouch and Beer, and Beer and Thera, as well as other related results, can be extended to normed spaces if subdifferentials are replaced by ε-subdifferentials. These extensions are possible because in [13] we proved that Borwein’s variational principle and Rockafellar’s maximal monotonicity theorem are true in normed spaces, provided that we replace subdifferentials with ε-subdifferentials.

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag Berlin

88 A. Verona, M.E. Verona / Epiconvergence and ε-subgradients of convex functions

2. Notation and preliminary results

In this first part we shall introduce the notation and recall several known results used in the paper. We shall also extend to ε-subgradients several results concerning subgradients. Let X be a normed space, B be its unit ball, X^{∗} be the topological dual of X, and B^{∗} be the unit ball in X^{∗} (with respect to the dual norm). On X^{∗} we consider only the strong topology and on products (such as X × X^{∗} or X × R) we consider the box norm. We write Γ(X) (resp. Γ(X^{∗})) for the set of all proper, lower semicontinuous, convex functions defined on X (resp. on X^{∗}) and with values in (−∞, +∞]. As usual, we denote a multivalued map by a double arrow. Given f ∈ Γ(X), x ∈ domf, and ε ≥ 0, we denote by ∂_{ε}f(x) the set of all ε-subgradients to f at x and by ∂_{ε}f the set of all pairs (x, x^{∗}) such that x^{∗} ∈ ∂_{ε}f(x). The conjugate f^{∗} ∈ Γ(X^{∗}) of f is defined by

f^{∗}(x^{∗}) = sup{⟨x^{∗}, x⟩ − f(x); x ∈ X}.

Then

⟨x^{∗}, x⟩ ≤ f^{∗}(x^{∗}) + f(x) for any x ∈ X, x^{∗} ∈ X^{∗} (Fenchel’s inequality)  (1)

and, if ε ≥ 0,

f^{∗}(x^{∗}) + f(x) ≤ ⟨x^{∗}, x⟩ + ε if and only if (x, x^{∗}) ∈ ∂_{ε}f.  (2)

Also

f(x) = sup{⟨x^{∗}, x⟩ − f^{∗}(x^{∗}); x^{∗} ∈ X^{∗}}.  (3)

The following Brøndsted-Rockafellar type result was proved in [13].

Proposition 2.1. Let f ∈ Γ(X), ε ≥ 0, δ > 0 (δ ≥ 0 if X is a Banach space), (x, x^{∗}) ∈ ∂_{ε+δ}f, and λ > 0. Then there exists (y, y^{∗}) ∈ ∂_{δ}f such that:

(a) ∥y − x∥ ≤ ε/λ;

(b) ∥y^{∗} − x^{∗}∥ ≤ λ;

(c) |f(y) − f(x)| ≤ ε(1 + \frac{∥x^{∗}∥}{λ}).

Definition 2.2.

(1) For f ∈ Γ(X), let ∂_{•}f denote the set of all triples (ε, x, x^{∗}) ∈ R × X × X^{∗} such that ε ≥ 0 and x^{∗} ∈ ∂_{ε}f(x).

(2) For f ∈ Γ(X), define d_{f} : domf × domf^{∗} → [0, +∞) by d_{f}(u, u^{∗}) = f^{∗}(u^{∗}) + f(u) − ⟨u^{∗}, u⟩. Thus u^{∗} ∈ ∂_{d_{f} (u,u^{∗})}f(u) but u^{∗} ∉ ∂_{ε}f(u) if ε < d_{f}(u, u^{∗}).

(3) For f ∈ Γ(X), let ∂_{♯}f denote the subset of all (ε, u, u^{∗}) ∈ ∂_{•}f with ε = d_{f}(u, u^{∗}).

Later we shall need the following approximation result.

Lemma 2.3. Let f ∈ Γ(X) and (ε, x^{∗}, x^{∗∗}) ∈ ∂_{♯}f^{∗}. Then there exist a bounded net {x_{ι}} in X, weak^{∗∗}-convergent to x^{∗∗}, and a net {ε_{ι}} converging to ε such that (ε_{ι}, x_{ι}, x^{∗}) ∈ ∂_{♯}f for each ι. A similar assertion with ∂_{•}f^{∗} instead of ∂_{♯}f^{∗} is also true.

Proof. Since epif is weak^{∗∗}-dense in epif^{∗∗} (see [12]), by Lemma D in [13], there exists a bounded net {x_{ι}} in X which is weak^{∗∗}-convergent to x^{∗∗} and such that the net {f(x_{ι})} converges to f^{∗∗}(x^{∗∗}). Take ε_{ι} = f^{∗}(x^{∗}) + f(x_{ι}) − ⟨x^{∗}, x_{ι}⟩.

As mentioned in the introduction, we shall need some facts from [13]. Recall that a family of multivalued maps (T_{ε})_{ε≥0} with T_{ε} : X ⇒ X^{∗} is called a monotone family of operators if for any x, y ∈ X, ε, δ ≥ 0, x^{∗} ∈ T_{ε}(x), and y^{∗} ∈ T_{δ}(y) we have ⟨x−y, x^{∗} −y^{∗}⟩ ≥ −ε−δ. A monotone family of operators (T_{ε})_{ε≥0} is called maximal if given any monotone family of operators (S_{ε})_{ε≥0} such that T_{ε}(x) ⊆ S_{ε}(x) for all x ∈ X and ε ≥ 0, then T_{ε}(x) = S_{ε}(x) for all x ∈ X and ε ≥ 0. One can check easily that (∂_{ε}f)_{ε≥0} is a monotone family of operators for any f ∈ Γ(X). In [13] we proved that such a family of monotone operators is maximal. The following result is a variant of the Rockafellar integration formula.

Lemma 2.4. Let f ∈ Γ(X) and (z, z^{∗}) ∈ domf × domf^{∗}. Then, for any x ∈ X,

f(x) − f(z) = sup {∑_{i=1}^{k} (⟨x_{i}^{∗}, x_{i−1} − x_{i}⟩ − d_{f}(x_{i}, x_{i}^{∗}))},

where x_{0} = x, x_{k} = z, x_{k}^{∗} = z^{∗} and the supremum is taken with respect to all k ≥ 2, (x_{i}, x_{i}^{∗}) ∈ domf × domf^{∗}, i = 1, …, k − 1.

Proof. Let

g(x) = sup {∑_{i=1}^{k}(⟨x_{i}^{∗}, x_{i−1} − x_{i}⟩ − ε_{i})} ,

where x_{0} = x, x_{k} = z, x_{k}^{∗} = z^{∗}, ε_{k} = d_{f}(z, z^{∗}) and the supremum is taken with respect to all k ≥ 2, (ε_{i}, x_{i}, x_{i}^{∗}) ∈ ∂_{•}f, i = 1, …, k − 1. It is easy to see that g(x) is also equal to the right side of the formula we want to prove. It was shown in [13] that g ∈ Γ(X), g(z) ≤ 0, ∂_{α}g = ∂_{α}f for any α ≥ 0, and that g and f differ by a constant. Thus the assertion of the lemma will follow if we prove that g(z) = 0. This is trivial if d_{f}(z, z^{∗}) = 0. So assume that ε = d_{f}(z, z^{∗}) > 0 and g(z) < 0. Then g(z) ≤ −β < 0 for some 0 < β < ε. For any δ ≥ 0 and any y^{∗} ∈ ∂_{δ}f(y) we have

−β ≥ g(z) ≥ ⟨y^{∗}, z − y⟩ − δ + ⟨z^{∗}, y − z⟩ − ε

implying that ⟨y^{∗} − z^{∗}, y − z⟩ ≥ −δ − (ε − β). From the maximal monotonicity of the family (∂_{δ}f) (see [13], Theorem 2.2) it follows that z^{∗} ∈ ∂_{ε−β}f(z), which contradicts the fact that ε = d_{f}(z, z^{∗}). Thus g(z) = 0 and the lemma is proved.

Remark 2.5. We shall also need the following dual result. Let f, ε, z, and z^{∗} be as in the previous lemma and let x^{∗} ∈ X^{∗}. Then

f^{∗}(x^{∗}) − f^{∗}(z^{∗}) = sup {∑_{i=1}^{k}(⟨x_{i−1}^{∗} − x_{i}^{∗}, x_{i}⟩ − d_{f}(x_{i}, x_{i}^{∗}))},

where x_{0}^{∗} = x^{∗}, x_{k} = z, x_{k}^{∗} = z^{∗} and the supremum is taken with respect to all k ≥ 2, (x_{i}, x_{i}^{∗}) ∈ domf × domf^{∗}, i = 1, …, k − 1. If we apply Lemma 2.4 to f^{∗}, we obtain the above formula, but with the supremum taken with respect to all couples (x_{i}^{∗}, x_{i}^{∗∗}) ∈ domf^{∗} × domf^{∗∗}. To obtain our result it is enough to use Lemma 2.3.

For any real number λ > 0 we denote by K_{λ} the closed, convex cone {(x, t) ∈ X × R; t ≤ −λ∥x∥}; the interior of K_{λ} is denoted K̊_{λ}. For λ > 0, the Lipschitz regularization with parameter λ of f is defined as follows (see for example [11]):

f_{λ}(x) = inf{f(u) + λ∥x − u∥; u ∈ X}.

Here are some of the properties of f_{λ} we need:

f_{λ}(x) ≤ f_{μ}(x) if λ ≤ μ; f(x) = lim_{λ→∞} f_{λ}(x) for anyx ∈ X;  (4)

\begin{matrix}either f_{λ}(x) = −∞ for all x ∈ X, or f_{λ} is Lipschitz on Xwith Lipschitz constant λ; \\ the second alternative occurs iff there exist ε ≥ 0 and (x, x^{∗}) ∈ ∂_{ε}f with∥x^{∗}∥ ≤ λ. \end{matrix} \}

(5)

For (x, x^{∗}) as above we have

f_{λ}(z) ≥ ⟨x^{∗}, z − x⟩ + f(x) − ε, for any z ∈ X.  (6)

If f_{λ} ≠ −∞ and x ∈ X, one can look at f_{λ}(x) as the largest real number t such that epif and the interior of K_{λ} + (x, t) are disjoint, i.e.

f_{λ}(x) = max {t; (K̊_{λ} + (x, t)) ∩ epif = ∅} = sup {t; (K_{λ} + (x, t)) ∩ epif = ∅} . (7)

Following the terminology introduced in [10], an ε-estimator for f_{λ}(z) is a point x ∈ X such that

f(x) + λ∥z − x∥ < f_{λ}(z) + ε.  (8)

Lemma 2.6. Let f ∈ Γ(X), f(0) = 0. Let r > 1, z ∈ rB, 0 < ε < 1, 0 ≤ δ_{0} < 1, x_{0}^{∗} ∈ ∂_{δ_{0}}f(0), λ_{0} = ∥x_{0}^{∗}∥(1 + 2r) + 2, and λ ≥ λ_{0}. Then:

(a) There exists (x, x^{∗}) ∈ ∂_{ε}f such that:

(i) x is an ε-estimator for f_{λ}(z);  (ii) ∥x^{∗}∥ ≤ λ;
(iii) ∥z − x∥ ≤ r + 1 (hence ∥x∥ ≤ 2r + 1);  (iv) |f(x)| ≤ λr + 1;
(v) f_{λ}(z) − f(x) ≤ ⟨x^{∗}, z − x⟩.

(b) The function λ ↦ f_{λ}(z) from [λ_{0}, ∞) to R is Lipschitz with Lipschitz constant r + 1.

Proof. From (5) it follows that f_{λ} is a Lipschitz function on X for any λ ≥ ∥x_{0}^{∗}∥. From (7), there exist x^{∗} ∈ X^{∗} and t ∈ R such that the graph of the affine functional x^{∗} + t separates epif and K_{λ} + (z, f_{λ}(z)). Then ∥x^{∗}∥ ≤ λ and for any x

f_{λ}(z) − λ∥z − x∥ ≤ ⟨x^{∗}, x⟩ + t ≤ f(x).

Choose now x to be an ε-estimator for f_{λ}(z). It follows from (8) and the above inequalities that 0 ≤ f(x) − (⟨x^{∗}, x⟩ + t) ≤ ε, hence x^{∗} ∈ ∂_{ε}f(x). Using (8) again we obtain

\begin{aligned}λ∥z − x∥ & ≤ f_{λ}(z) + ε − f(x) ≤ f(0) + λ∥z∥ + ε − ⟨x_{0}^{∗}, x⟩ + δ_{0} \\ & ≤ λ∥z∥ + ε + ⟨x_{0}^{∗}, z − x⟩ − ⟨x_{0}^{∗}, z⟩ + δ_{0} ≤ λr + ε + ∥x_{0}^{∗}∥∥z − x∥ + ∥x_{0}^{∗}∥r + δ_{0} \end{aligned}

hence

∥z − x∥ ≤ \frac{r(λ + ∥x_{0}^{∗}∥) + ε + δ_{0}}{λ − ∥x_{0}^{∗}∥}.

from which (iii) follows. To prove (iv) it remains to notice that

−λ ≤ −∥x_{0}^{∗}∥(2r + 1) − 2 ≤ ⟨x_{0}^{∗}, x⟩ − δ_{0} ≤ f(x) ≤ f_{λ}(z) + ε ≤ f(0) + λ∥z∥ + ε ≤ λr + 1.

To prove (v), it is enough to see (by what precedes) that

f_{λ}(z) ≤ ⟨x^{∗}, z⟩ + t = ⟨x^{∗}, z − x⟩ + ⟨x^{∗}, x⟩ + t ≤ ⟨x^{∗}, z − x⟩ + f(x).

To prove (b) let μ > ν ≥ λ_{0}, ε > 0, and x be an ε-estimator for f_{ν}(z). We have

0 ≤ f_{μ}(z) − f_{ν}(z) ≤ f(x) + μ∥z − x∥ − f(x) − ν∥z − x∥ + ε = (μ − ν)∥z − x∥ + ε.

By part (i), ∥z − x∥ ≤ r + 1, hence

|f_{μ}(z) − f_{ν}(z)| ≤ (μ − ν)(r + 1) + ε.

Since ε was arbitrary, this proves part (b).

3. Slice convergence and Kuratowski–Painlevé convergence of ε-subdifferen-tials

If X is a Banach space, a result of Attouch and Beer [2] (see also [1, Theorem 3.66]) links slice convergence of epigraphs of convex functions with the Kuratowski-Painlevé convergence of their subdifferentials. In this section we prove that a similar result is true in a normed space, provided that we replace subdifferentials with ε-subdifferentials. The main ideas of the proof are the same as in [2], with some changes due to the fact that we work with ε-subdifferentials. We begin by recalling the necessary notions.

Given a topological space T and a sequence {C_{n}} of nonempty subsets of T we denote by Li C_{n} the set of those x ∈ T such that there exists a sequence {x_{n}} converging to x, with x_{n} ∈ C_{n} for all n and denote by Ls C_{n} the set of those x ∈ T such that there exists a sequence 0 < n_{1} < n_{2} < … and a sequence {x_{k}} converging to x, with x_{k} ∈ C_{n_{k}} for all k. The sequence {C_{n}} is said to be Kuratowski-Painlevé convergent if Li C_{n} = Ls C_{n}; this common value is denoted Lim C_{n}. Another notion we need is that of slice topology on Γ(X), denoted τ_{S} (see [7], [9]). Recall that a subbase for τ_{S} consists of

all sets of the form (V ×(−∞, t))^{−}, with V open in X and t ∈ R, where (V ×(−∞, t))^{−}

is the set of all f ∈ Γ(X) such that epif ∩ (V × (−∞, t)) ≠ ∅;

and

all sets of the form (s(μ, x^{∗}, η)^{c})^{++}, with μ > 0, x^{∗} ∈ X^{∗}, and η ∈ R, where s(μ, x^{∗}, η) = {(x, t) ∈ X × R; ∥x∥ ≤ μ, t = ⟨x^{∗}, x⟩ − η} and (s(μ, x^{∗}, η)^{c})^{++} is the set of all f ∈ Γ(X) such that ((epif) + ε(B × [−1, 1])) ∩ s(μ, x^{∗}, η) = ∅ for some ε > 0.

The following characterization of convergence in this topology is an adaptation to the context of normed spaces of Theorem 3.1 in [2].

Proposition 3.1. Let X be a normed space, f, f_{n} ∈ Γ(X), n ≥ 1. The following are equivalent:

(i) f = τ_{S}-limf_{n}; (ii) for any (x, x^{∗}) ∈ X × X^{∗} there exist a sequence {x_{n}} in X and a sequence {x_{n}^{∗}} in

X^{∗} converging to x and to x^{∗} respectively and such that f(x) = lim_{n→∞} f_{n}(x_{n}), and

f^{∗}(x^{∗}) = lim_{n→∞} f_{n}^{∗}(x_{n}^{∗}).

(iii) for any (x, x^{∗}) ∈ domf × domf^{∗} and any α > 0 there exist a sequence {x_{n}} in X and a sequence {x_{n}^{∗}} in X^{∗} converging to x and to x^{∗} respectively and such that

f(x) + α ≥ lim sup_{n→∞} f_{n}(x_{n}) and f^{∗}(x^{∗}) + α ≥ lim sup_{n→∞} f_{n}^{∗}(x_{n}^{∗}).

Proof. The fact that (i) implies (ii) is proved in [2], Theorem 3.1. Clearly (ii) implies (iii). It remains to show that (iii) implies (i). So assume that (iii) is true, but (i) is false. Then, by restricting our attention to a subsequence, we can assume that either

(a) there exist an open subset V of X and t ∈ R such that epif ∩ (V × (−∞, t)) ≠ ∅ but epif_{n} ∩ (V × (−∞, t)) = ∅ for all n

or

(b) there exit μ > 0, z^{∗} ∈ X^{∗}, η ∈ R and ε > 0 such that ((epif) + ε(B × [−1, 1])) ∩ s(μ, z^{∗}, η) = ∅ and ((epif_{n}) + ε_{n}(B × [−1, 1])) ∩ s(μ, z^{∗}, η) ≠ ∅ for any ε_{n} > 0 and any n;

is true. It is not difficult to see that, because of (iii) (the part involving x), (a) cannot be true. So (b) must be true. From Lemma 8.1.1 in [9] it follows that there exists (x^{∗}, t) ∈ epif^{∗} such that the graph of the affine functional x^{∗} − t strongly separates epif and s(μ, z^{∗}, η), i.e.

⟨x^{∗}, u⟩ − t ≤ f(u), u ∈ X

and

⟨z^{∗}, u⟩ − η ≤ ⟨x^{∗}, u⟩ − t − δ, ∥u∥ ≤ μ.

for some δ > 0. It is easy to see that there exists μ′ > μ such that the last inequality becomes

⟨z^{∗}, u⟩ − η ≤ ⟨x^{∗}, u⟩ − t − δ/2, ∥u∥ ≤ μ′.

By hypothesis there exists a sequence {x_{n}^{∗}} in X^{∗} converging to x^{∗} and such that f^{∗}(x^{∗})+ δ/16 ≥ lim sup_{n→∞} f_{n}^{∗}(x_{n}^{∗}). Choose n large enough such that ∥x^{∗}−x_{n}^{∗}∥ < δ/(8μ′) and f^{∗}(x^{∗})+

δ/8 > f_{n}^{∗}(x_{n}^{∗}). Let u ∈ μ′B. By Fenchel’s inequality, our choices, and the previous inequalities we have

f_{n}(u) ≥ ⟨x_{n}^{∗}, u⟩ − f_{n}^{∗}(x_{n}^{∗}) = ⟨x_{n}^{∗} − x^{∗}, u⟩ + ⟨x^{∗}, u⟩ − f_{n}^{∗}(x_{n}^{∗})
≥ −δ/8 + ⟨x^{∗}, u⟩ − f^{∗}(x^{∗}) − δ/8 ≥ ⟨x^{∗}, u⟩ − t − δ/4
≥ ⟨z^{∗}, u⟩ − η + δ/4.

The resulting inequality shows that, if ε_{n} > 0, ε_{n} < μ′ − μ and 4ε_{n}(1 + ∥z^{∗}∥) < δ, then (epif_{n} + ε_{n}(B × [−1, 1])) ∩ s(μ, z^{∗}, η) = ∅, which contradicts (b). It follows that (iii) implies (i) and the proposition is proved.

Let now f, f_{n} ∈ Γ(X), n ≥ 1. Recall that a sequence {(x_{n}, x_{n}^{∗})} with (x_{n}, x_{n}^{∗}) ∈ domf_{n} ×

domf_{n}^{∗} is called normalizing if lim_{n→∞}(x_{n}, x_{n}^{∗}) = (x, x^{∗}) ∈ domf × domf^{∗}, lim_{n→∞} f_{n}(x_{n}) =

f(x), and lim_{n→∞} f_{n}^{∗}(x_{n}^{∗}) = f^{∗}(x^{∗}).

Lemma 3.2. Let f, f_{n} ∈ Γ(X), n ≥ 1. Let also {(x_{n}, x_{n}^{∗})} be a normalizing sequence and let lim_{n→∞}(x_{n}, x_{n}^{∗}) = (x, x^{∗}). Then

(1) lim_{n→∞} d_{f_{n}}(x_{n}, x_{n}^{∗}) = d_{f}(x, x^{∗}).

(2) Assume that ∂_{ε}f ⊆ Li ∂_{ε}f_{n} for every ε > 0. Let (u, u^{∗}) ∈ domf × domf^{∗} and

(u_{n}, u_{n}^{∗}) ∈ X × X^{∗}, n ≥ 1, be such that lim_{n→∞}(u_{n}, u_{n}^{∗}) = (u, u^{∗}). Let α = d_{f}(u, u^{∗}) and

α_{n} = d_{f_{n}}(u_{n}, u_{n}^{∗}). Then

lim sup_{n→∞} f_{n}(u_{n}) + α − lim sup_{n→∞} α_{n} ≤ f(u) ≤

lim inf_{n→∞} f_{n}(u_{n}) + lim sup_{n→∞} α_{n} − lim inf_{n→∞} α_{n}

and

lim sup_{n→∞} f_{n}^{∗}(u_{n}^{∗}) + α − lim sup_{n→∞} α_{n} ≤ f^{∗}(u^{∗}) ≤

lim inf_{n→∞} f_{n}^{∗}(u_{n}^{∗}) + lim sup_{n→∞} α_{n} − lim inf_{n→∞} α_{n}

(3) If in addition to the conditions in (2) we have lim sup_{n→∞} d_{f_{n}}(u_{n}, u_{n}^{∗}) ≤ d_{f}(u, u^{∗}), then

{(u_{n}, u_{n}^{∗})} is a normalizing sequence, i.e. lim_{n→∞} f_{n}(u_{n}) = f(u) and lim_{n→∞} f_{n}^{∗}(u_{n}^{∗}) =

f^{∗}(u^{∗}).

94 A. Verona, M.E. Verona / Epiconvergence and ε-subgradients of convex functions

Proof. The proof of (1) is trivial, so we shall prove only (2) and (3). Let γ = lim inf_{n→∞} α_{n},

and β = lim sup_{n→∞} α_{n}. Let k ≥ 2 and (ε_{i}, y_{i}, y_{i}^{∗}) ∈ ∂_{•}f, i = 1, …, k − 1. Let μ > 0 and μ_{i} =

ε_{i} + μ/(k − 1) (this is needed in case ε_{i} = 0). Then (μ_{i}, y_{i}, y_{i}^{∗}) ∈ ∂_{•}f, i = 1, …, k − 1 and

since ∂_{μ_{i}}f ⊆ Li ∂_{μ_{i}}f_{n}, there exist (y_{i,n}, y_{i,n}^{∗}) ∈ ∂_{μ_{i}}f_{n} such that lim_{n→∞}(y_{i,n}, y_{i,n}^{∗}) = (y_{i}, y_{i}^{∗}),

i = 1, …, k − 1. Let also y_{0} = x, y_{0,n} = x_{n}, y_{k} = u, y_{k}^{∗} = u^{∗}, ε_{k} = α, y_{k,n} = u_{n}, and y_{k,n}^{∗} = u_{n}^{∗}, (thus lim sup_{n→∞} α_{n} = β = ε_{k} − α + β). From the definition of ε-subgradients it follows that

f_{n}(x_{n}) ≥ f_{n}(y_{k,n}) + ∑_{i=1}^{k}⟨y_{i,n}^{∗}, y_{i−1,n} − y_{i,n}⟩ − ∑_{i=1}^{k−1} μ_{i} − α_{n}, n ≥ 1.

By replacing μ_{i} with ε_{i} + μ/(k − 1) and taking lim sup (with respect to n) we obtain

f(x) ≥ lim sup_{n→∞} f_{n}(u_{n}) + ∑_{i=1}^{k}⟨y_{i}^{∗}, y_{i−1} − y_{i}⟩ − ∑_{i=1}^{k} ε_{i} + α − β − μ.

Lemma 2.4 (more exactly its proof) implies that

f(x) ≥ lim sup_{n→∞} f_{n}(u_{n}) + f(x) − f(u) + α − β − μ

and, since μ is arbitrary,

f(u) ≥ lim sup_{n→∞} f_{n}(u_{n}) + α − β.

Similar arguments (the remark following Lemma 2.4 is essential) show that

f^{∗}(u^{∗}) ≥ lim sup_{n→∞} f_{n}^{∗}(u_{n}^{∗}) + α − β.

Since

f_{n}^{∗}(u_{n}^{∗}) + f_{n}(u_{n}) = ⟨u_{n}^{∗}, u_{n}⟩ + α_{n},

it follows that

lim inf_{n→∞} f_{n}(u_{n}) ≥ ⟨u^{∗}, u⟩ + γ − lim sup_{n→∞} f_{n}^{∗}(u_{n}^{∗})
≥ ⟨u^{∗}, u⟩ + α − f^{∗}(u^{∗}) + γ − β = f(u) + γ − β.

and therefore

f(u) ≤ lim inf_{n→∞} f_{n}(u_{n}) + β − γ.

The corresponding inequality involving the conjugate functions can be proved in the same way. This proves (2). To prove (3), assume first that β = γ ≤ α. From (2) it follows that β = α, lim_{n→∞} f_{n}(u_{n}) = f(u), and lim_{n→∞} f_{n}^{∗}(u_{n}^{∗}) = f^{∗}(u^{∗}). To prove the general case (β ≤ α), notice that the above arguments imply that every convergent subsequence of {α_{n}} must converge to α. Therefore the sequence {α_{n}} is convergent (to α) and the lemma is proved.

Lemma 3.3. Let f, f_{n} ∈ Γ(X), n ≥ 1. Then

(1) ∂_{•}f ⊆ Li ∂_{•}f_{n} if and only if ∂_{•}f = Lim ∂_{•}f_{n}

(2) Given any ε ≥ 0, ∂_{ε}f ⊆ Li ∂_{ε}f_{n} if and only if ∂_{ε}f = Lim ∂_{ε}f_{n}.

Proof. We shall prove only the first assertion, the proof of the other one being similar. Since trivially Li ∂_{•}f_{n} ⊆ Ls ∂_{•}f_{n}, it is enough to prove that Ls ∂_{•}f_{n} ⊆ ∂_{•}f if ∂_{•}f ⊆ Li ∂_{•}f_{n}. To this end let (α, x, x^{∗}) ∈ Ls ∂_{•}f_{n}. By definition there exist an increasing sequence of positive integers n_{1} < n_{2} < … < n_{k} < … and a sequence {(α_{k}, x_{k}, x_{k}^{∗})} with (α_{k}, x_{k}, x_{k}^{∗}) ∈ ∂_{•}f_{n_{k}} such that

lim_{k→∞}(α_{k}, x_{k}, x_{k}^{∗}) = (α, x, x^{∗}).

Let ε ≥ 0 and let (y, y^{∗}) ∈ ∂_{ε}f. Then (ε, y, y^{∗}) ∈ ∂_{•}f and, since ∂_{•}f ⊆ Li ∂_{•}f_{n}, there exist (ε_{n}, y_{n}, y_{n}^{∗}) ∈ ∂_{•}f_{n}, n ≥ 1, such that lim_{n→∞}(ε_{n}, y_{n}, y_{n}^{∗}) = (ε, y, y^{∗}). It is easy to see that

⟨y_{n_{k}}^{∗} − x_{k}^{∗}, y_{n_{k}} − x_{k}⟩ ≥ −ε_{n_{k}} − α_{k}

and, by taking the limit, we obtain

⟨y^{∗} − x^{∗}, y − x⟩ ≥ −ε − α.

The fact that (∂_{ε}f)_{ε} is a maximal family of monotone operators (see [13], Theorem 2) implies that x^{∗} ∈ ∂_{α}f(x), showing that Ls ∂_{•}f_{n} ⊆ ∂_{•}f.

We can now state and prove our main result in this section.

Theorem 3.4. Let X be a normed space and f, f_{n} ∈ Γ(X), n ≥ 1. The following assertions are equivalent:

(1) f = τ_{S}-limf_{n};

(2) ∂_{♯}f ⊆ Li ∂_{♯}f_{n} and there exists a normalizing sequence;

(3) ∂_{•}f = Lim ∂_{•}f_{n} and there exists a normalizing sequence;

(4) ∂_{ε}f = Lim ∂_{ε}f_{n} for every ε > 0 and there exists a normalizing sequence.

Proof. First we shall prove that (1) implies (2). To this end assume that (1) is true and let (ε, x, x^{∗}) ∈ ∂_{♯}f. It follows that there exist sequences {x_{n}} and {x_{n}^{∗}} such that

x = lim_{n→∞} x_{n}, x^{∗} = lim_{n→∞} x_{n}^{∗} and f(x) = lim_{n→∞} f_{n}(x_{n}), f^{∗}(x^{∗}) = lim_{n→∞} f_{n}^{∗}(x_{n}^{∗}). Let ε_{n} =

f_{n}^{∗}(x_{n}^{∗}) + f_{n}(x_{n}) − ⟨x_{n}^{∗}, x_{n}⟩, n ≥ 1. Then (ε_{n}, x_{n}, x_{n}^{∗}) ∈ ∂_{♯}f_{n} and clearly lim_{n→∞} ε_{n} = ε. Thus

(ε, x, x^{∗}) ∈ Li ∂_{♯}f_{n} and therefore ∂_{♯}f ⊆ Li ∂_{♯}f_{n}.

Assume now that (2) is true and let (ε, u, u^{∗}) ∈ ∂_{•}f. Then (α, u, u^{∗}) ∈ ∂_{♯}f, where α = d_{f}(u, u^{∗}). Since (2) is assumed true, there exist (α_{n}, u_{n}, u_{n}^{∗}) ∈ ∂_{♯}f_{n}, n ≥ 1, such

that lim_{n→∞}(α_{n}, u_{n}, u_{n}^{∗}) = (α, u, u^{∗}). Then (ε + α_{n} − α, u_{n}, u_{n}^{∗}) ∈ ∂_{•}f_{n} and lim_{n→∞}(ε + α_{n} −

α, u_{n}, u_{n}^{∗}) = (ε, u, u^{∗}), showing that ∂_{•}f ⊆ Li ∂_{•}f_{n}. From Lemma 3.3 it follows that (3) is true. Next we shall show that (3) implies (4). So assume that (3) is true and let ε > 0 and u^{∗} ∈ ∂_{ε}f(u). Our assumption implies that there exist (ε_{n}, u_{n}, u_{n}^{∗}) ∈ ∂_{•}f_{n}, n ≥ 1, such that lim_{n→∞}(ε_{n}, u_{n}, u_{n}^{∗}) = (ε, u, u^{∗}). Let δ_{n} = min{ε, ε_{n}}. If δ_{n} < ε_{n}, we can use Proposition

2.1 to find (v_{n}, v_{n}^{∗}) ∈ ∂_{δ_{n}}f_{n} such that ∥v_{n} − u_{n}∥ ≤ \sqrt{ε_{n} − δ_{n}} and ∥v_{n}^{∗} − u_{n}^{∗}∥ ≤ \sqrt{ε_{n} − δ_{n}}. If δ_{n} = ε_{n}, take v_{n} = u_{n} and v_{n}^{∗} = u_{n}^{∗}. It follows that lim_{n→∞} v_{n} = u and lim_{n→∞} v_{n}^{∗} = u^{∗} and,

since δ_{n} ≤ ε, ∂_{δ_{n}}f_{n} ⊆ ∂_{ε}f_{n}. Thus ∂_{ε}f ⊆ Li ∂_{ε}f_{n}. From Lemma 3.3 it follows that (4) is true. Finally assume that (4) is true. In order to prove that (1) is true, it will be enough to show that for any (u, u^{∗}) ∈ domf × domf^{∗} and any μ > 0 there exist a sequence {u_{n}} in X converging to u and a sequence {u_{n}^{∗}} in X^{∗} converging to u^{∗} such that f(u) + μ ≥ lim sup_{n→∞} f_{n}(u_{n}) and f^{∗}(u^{∗}) + μ ≥ lim sup_{n→∞} f_{n}^{∗}(u_{n}^{∗}) (see Proposition 3.1). To this end, let

u ∈ domf, u^{∗} ∈ domf^{∗}, μ > 0, and ε = d_{f}(u, u^{∗}). Then (u, u^{∗}) ∈ ∂_{ε+μ}f and, by hypothesis, there exists a sequence {(u_{n}, u_{n}^{∗})} with (u_{n}, u_{n}^{∗}) ∈ ∂_{ε+μ}f_{n} and such that lim_{n→∞}(u_{n}, u_{n}^{∗}) = (u, u^{∗}). By Lemma 3.2, {u_{n}} and {u_{n}^{∗}} have the required properties.

The following example shows that we cannot expect equality in (2) above.

Example 3.5. Define f_{n} ∈ Γ(R) by f_{n}(x) = −1 if x > 1/n and f_{n}(x) = −nx if 0 ≤ x ≤ 1/n. Then τ_{S}-limf_{n} = f, where f ∈ Γ(R) is given by f(x) = −1 if x ≥ 0. Clearly (1, 0, 0) ∈ ∂_{♯}f_{n} for all n ≥ 1, so (1, 0, 0) ∈ Li ∂_{♯}f_{n}. However, d_{f}(0, 0) = 0, so (1, 0, 0) is not an element of ∂_{♯}f.

Remark 3.6. If X is Banach, it was proved in [2] that assertion (1) of Theorem 3.4 is equivalent to

∂f = Lim ∂f_{n} and there exists a normalizing sequence.

The following characterization of the slice topology was proved in [7], Theorem 4.11: If X is a Banach space, the slice topology on Γ(X) is the weakest topology on Γ(X) for which the multifunction Δ : Γ(X) ⇒ X × R × X^{∗}, Δ(f) = {(x, f(x), x^{∗}); (x, x^{∗}) ∈ ∂f} is lower semicontinuous. In the case of normed spaces we have:

Theorem 3.7. Let X be a normed space.

(1) The slice topology is the weakest topology on Γ(X) for which all multifunctions Δ_{ε} : Γ(X) ⇒ X × R × X^{∗}, Δ_{ε}(f) = {(x, f(x), x^{∗}); (x, x^{∗}) ∈ ∂_{ε}f}, ε > 0, are lower semicontinuous.

(2) The slice topology is the weakest topology on Γ(X) for which the multifunction Δ_{♯} : Γ(X) ⇒ X × R × X^{∗} × R, Δ_{♯}(f) = {(x, f(x), x^{∗}, ε); x ∈ domf, x^{∗} ∈ domf^{∗}, ε = d_{f}(x, x^{∗})}, is lower semicontinuous.

Proof. Recall first that a multifunction T : A ⇒ B, with A and B topological spaces, is lower semicontinuous if for any net {a_{ι}} converging to a in A we have T (a) ⊆ Li T (a_{ι}). Notice next that in all our results we can replace sequences with nets. Now, arguments similar to those used in the proof of Theorem 3.4 show that the multifunctions Δ_{♯} and Δ_{ε} are lower semicontinuous when Γ(X) is endowed with the slice topology. Let τ be a topology on Γ(X) for which all multifunctions Δ_{ε} are lower semicontinuous. In order to prove that the slice topology is weaker than τ it is enough to show that if a net {f_{ι}} is τ-convergent to f then it is also τ_{s}-convergent to f. To this end, let {f_{ι}} be such a net, (x, x^{∗}) ∈ domf × domf^{∗}, ε = d_{f}(x, x^{∗}), and μ > 0. The τ-lower semicontinuity of Δ_{ε+μ} implies that Δ_{ε+μ}(f) ⊆ Li Δ_{ε+μ}(f_{ι}). Since (x, f(x), x^{∗}) ∈ Δ_{ε+μ}(f), there exists a net (x_{ι}, f_{ι}(x_{ι}), x_{ι}^{∗}) ∈ Δ_{ε+μ}(f_{ι}) converging to (x, f(x), x^{∗}). From this we get that lim f_{ι}(x_{ι}) = f(x) and lim sup f_{ι}^{∗}(x_{ι}^{∗}) ≤ f^{∗}(x^{∗}) + μ. Proposition 3.1 implies that {f_{ι}} τ_{s}-converges to f and this completes the proof of (1). The proof of (2) is similar.

4. Attouch–Wets convergence and ε-subdifferentials

Another useful topology on the set of closed, convex sets is that introduced by Attouch and Wets [4]; if convex functions are identified with their epigraphs, we obtain the Attouch-Wets topology on Γ(X), denoted τ_{AW}. In [10], Theorem 3.6, this topology is characterized in terms of the “uniform lower semicontinuity on bounded sets” of the multifunction Δ : Γ(X) ⇒ X × R × X^{∗} (X is assumed to be Banach). In this section we show that a similar characterization is true in normed spaces, provided that we replace subdifferentials with ε-subdifferentials. The ideas of the proof are those in [10], with changes due to the fact that we work with ε-subdifferentials. We begin by recalling the necessary notions (for details see [4], [7], [5]). Let X be a normed space. If A, C are subsets of X, the excess of C over A is defined by

e(C, A) = sup_{c∈C} inf_{a∈A} ∥c − a∥ = sup_{c∈C} d(c, A).

For ρ > 0 we write e_{ρ}(C, A) = e(C ∩ ρB, A). The ρ-Hausdorff distance between A and C is

haus_{ρ}(A, C) = max{e_{ρ}(A, C), e_{ρ}(C, A)}.

Finally we say that a sequence {f_{n}} of functions from Γ(X) converges to f ∈ Γ(X) in the Attouch-Wets sense, denoted τ_{AW}-limf_{n} = f, if lim_{n→∞} haus_{ρ}(epif_{n}, epif) = 0 for each

ρ > 0 (in fact, it is enough that this happens for any ρ larger than some positive number).

Lemma 4.1. Let 0 < ε < 1/2, δ > 0 (δ ≥ 0 if X is Banach), and r > max{2, δ}. If f, g ∈ Γ(X) verify e_{r}(epif, epig) < ε^{2} and e_{2r^{2}}(epif^{∗}, epig^{∗}) < ε^{2} then e_{r}(Δ_{δ}(f), Δ_{δ}(g)) ≤ δ + 6r^{2}ε.

Proof. Let (x, f(x), x^{∗}) ∈ Δ_{δ}(f) ∩ (rB × [−r, r] × rB^{∗}). Then |f^{∗}(x^{∗})| ≤ 2r^{2} and there exist (y, t) ∈ epig and (y^{∗}, t^{∗}) ∈ epig^{∗} such that

∥y − x∥ ≤ ε^{2}, |t − f(x)| ≤ ε^{2}, ∥y^{∗} − x^{∗}∥ ≤ ε^{2}, |t^{∗} − f^{∗}(x^{∗})| ≤ ε^{2}.

It follows that

g(y) − f(x) ≤ ε^{2}, g^{∗}(y^{∗}) − f^{∗}(x^{∗}) ≤ ε^{2}, ezrm and |⟨y^{∗}, y⟩ − ⟨x^{∗}, x⟩| ≤ 3rε^{2}.

The above inequalities give

\begin{aligned}g^{∗}(y^{∗}) + g(y) & ≤ f^{∗}(x^{∗}) + f(x) + 2ε^{2} ≤ ⟨x^{∗}, x⟩ + δ + 2ε^{2} \\ & ≤ ⟨y^{∗}, y⟩ + δ + 3rε^{2} + 2ε^{2} ≤ ⟨y^{∗}, y⟩ + δ + 4r^{2}ε^{2} \end{aligned}

and therefore y^{∗} ∈ ∂_{δ+4r^{2}ε^{2}}g(y). From Proposition 2.1 it follows that there exist (z, z^{∗}) ∈ ∂_{δ}g such that

∥z − y∥ ≤ 2rε, ∥z^{∗} − y^{∗}∥ ≤ 2rε, ezrm and |g(z) − g(y)| ≤ 4r^{2}ε.

We also have

f(x) − g(y) ≤ ⟨x^{∗}, x⟩ − f^{∗}(x^{∗}) + δ + g^{∗}(y^{∗}) − ⟨y^{∗}, y⟩ ≤ 3rε^{2} + ε^{2} + δ ≤ 2r^{2}ε + δ.

Thus

∥z − x∥ ≤ 3rε, ∥z^{∗} − x^{∗}∥ ≤ 3rε, and |g(z) − f(x)| ≤ δ + 6r^{2}ε,

meaning that e_{r}(Δ_{δ}(f), Δ_{δ}(g)) ≤ δ + 6r^{2}ε.

Lemma 4.2. Let f ∈ Γ(X), f(0) = 0 and choose λ_{0} as in Lemma 2.6. Let also 0 < ε < 1, 0 < δ < 1, λ ≥ λ_{0}, and ρ = max{2r + 1, λ, λr + 1}. Assume that g ∈ Γ(X) satisfies e_{ρ}(Δ_{δ}(f), Δ_{δ}(g)) < ε + δ. Then sup{|f_{λ}(z) − g_{λ}(z)|; z ∈ rB} ≤ 5ρ(ε + δ).

Proof. Let z ∈ rB and let x be a δ-estimator for f_{λ}(z). Choose x^{∗} as in Lemma 2.6. Then (x, f(x), x^{∗}) ∈ Δ_{δ}(f)∩(ρB ×[−ρ, ρ]×ρB^{∗}) and therefore there exists (y, g(y), y^{∗}) ∈ Δ_{δ}(g) such that

∥x − y∥ ≤ ε + δ, |f(x) − g(y)| ≤ ε + δ, ∥x^{∗} − y^{∗}∥ ≤ ε + δ.

Using the definition of g_{λ}(z), the fact that x is a δ-estimator for f_{λ}(z), and the above inequalities, we have

\begin{aligned}g_{λ}(z) − f_{λ}(z) & ≤ g(y) + λ∥z − y∥ − f(x) − λ∥z − x∥ + δ \\ & ≤ ε + δ + λ∥x − y∥ + δ ≤ ε + 2δ + λ(ε + δ) ≤ 2ρ(ε + δ). \end{aligned}

From Lemma 2.6 (a) (v) it follows that f_{λ}(z) ≤ ⟨x^{∗}, z⟩ + f(x) − ⟨x^{∗}, x⟩. Notice also that ∥y^{∗}∥ ≤ λ + ε + δ. Using the previous estimation for f_{λ}(z) and (6), we have

f_{λ}(z) − g_{λ+ε+δ}(z) ≤ ⟨x^{∗}, z⟩ + f(x) − ⟨x^{∗}, x⟩ − ⟨y^{∗}, z − y⟩ − g(y) + δ
≤ f(x) − g(y) + ⟨x^{∗} − y^{∗}, z⟩ + ⟨y^{∗} − x^{∗}, y⟩ + ⟨x^{∗}, y − x⟩ + δ

≤ ε + δ + r(ε + δ) + (ε + δ)(ρ + ε + δ) + ρ(ε + δ) + δ ≤ 4ρ(ε + δ).

Finally, from this and Lemma 2.6 (b) we obtain that f_{λ}(z) − g_{λ}(z) ≤ 5ρ(ε + δ) which completes the proof of the lemma.

We can now prove the main result of this section.

Theorem 4.3. Let X be normed space and f, f_{n} ∈ Γ(X), n ≥ 1. The following statements are equivalent:

(1) f = τ_{AW}-limf_{n};

(2) for any ε > 0 there exists ρ_{0} > 0 such that lim sup_{n→∞} e_{ρ}(Δ_{ε}(f), Δ_{ε}(f_{n})) ≤ ε if ρ ≥ ρ_{0}.

Proof. The fact that (1) implies (2) follows immediately from Lemma 4.1 and a result established in [6] which asserts that the Fenchel transform (i.e. f ↦ f^{∗} from Γ(X) to Γ(X^{∗})) is continuous with respect to the Attouch-Wets topology. The other implication follows from Lemma 4.2 and another result of Beer [8, Theorem 4.3] which asserts that f = τ_{AW}-limf_{n} if and only if there exists λ_{0} such that, for any λ ≥ λ_{0}, the sequence {(f_{n})_{λ}} converges uniformly on bounded sets to f_{λ}.

Question. Is it possible to replace (2) in the above theorem with (2′) for any ε > 0 there exists ρ_{0} > 0 such that lim sup_{n→∞} e_{ρ}(Δ_{ε}(f), Δ_{ε}(f_{n})) = 0 if ρ ≥ ρ_{0}.

We conclude with an extension to normed spaces of a result from [3].

Theorem 4.4. Let X be normed space and f, f_{n} ∈ Γ(X), n ≥ 1. Assume that f = τ_{AW}- limf_{n}. Then, for any ε > 0, ∂_{ε}f = gph-dist lim_{n→∞} ∂_{ε}f_{n} (i.e. lim_{n→∞} haus_{ρ}(∂_{ε}f, ∂_{ε}f_{n}) = 0 for

ρ sufficiently large).

Proof. From the proof of Lemma 4.1 it follows that lim_{n→∞} e_{ρ}(∂_{ε}f, ∂_{ε}f_{n}) = 0 for ρ large

enough. The arguments given in the proof of Theorem 2.3 of [3] can be adapted to our context as in the cases discussed above and one can show that lim_{n→∞} e_{ρ}(∂_{ε}f_{n}, ∂_{ε}f) = 0 too.

Acknowledgement. We would like to thank L. Thibault for pointing out several errors in an earlier version of the paper and for some valuable suggestions.

References

[1] H. Attouch, Variational convergence for functions and operators, Applicable Math. Series, Pitman, London, 1984.

[2] H. Attouch, G. Beer, On the convergence of subdifferentials of convex functions, Arch. Math. 60 (1993), 389–400.

[3] H. Attouch, J. Ndoutoume, M. Thera, Epigraphical convergence of functions and convergence of their derivatives in Banach spaces, Sem. d’Anal. Convexe Montpellier 20 (1990), 9.1–9.45.

[4] H. Attouch, R. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), 695–730.

[5] D. Azé, J. P. Penot, Operations on convergent families of sets and functions, Optimization, 21 (1990), 521–534.

[6] G. Beer, Conjugate convex functions and the epi-distance topology, Proc. Amer. Math. Soc. 108 (1990), 117–126.

[7] G. Beer, The slice topology: A viable alternative to Mosco convergence in nonreflexive spaces, Nonlinear Analysis 19 (1992), 271–290.

[8] G. Beer, Lipschitz regularization and the convergence of convex functions, Num. Func. Anal. and Optimiz. 15 (1994), 31–46.

[9] G. Beer, Topologies on Closed and Closed Convex Sets, Kluwer Academic Publishers, 1993.

[10] G. Beer, M. Théra, Attouch-Wets convergence and a differential operator for convex functions, to appear in Proc. Amer. Math. Soc.

[11] J.-B. Hiriart-Urruty, Lipschitz r-continuity of the approximate subdifferential of a convex function. Math. Scand. 47 (1980), 123–134.

[12] J.-J. Moreau, Fonctionelles convexes, Lecture Notes (mimeographed), Collège de France, 1967.

[13] A. Verona, M. E. Verona, Remarks on subgradients and ε-subgradients, Set-Valued Analysis, 1 (1993), 261–272