Journal of Convex Analysis Volume 1 (1994), No.1, 101–105

An Attempt of Characterization of Functions With Sharp Weakly Complete Epigraphs

Jean Saint-Pierre, Michel Valadier

Département de Mathématiques, Case 051, Université Montpellier II, Place Eugène Bataillon, F-34095 Montpellier Cedex 5

We attempt to characterize the property, for a convex function f, “Epif is sharp and weakly complete” by a property of the polar function. Some examples show that a result completely analogous to the J.L. Joly characterization of “Epif is sharp and weakly locally compact” seems impossible. We use an old (1962) result of G. Choquet.

Keywords : Weakly complete sets, sharp epigraphs, locally convex linear spaces, duality

1991 Mathematics Subject Classification: 46A20, 46A55

1. Introduction

G. Choquet studied in 1962 ([4]–[6]) the weakly complete convex subsets of a Hausdorff locally convex topological vector space (briefly lctvs). Those which are sharp (i.e. which contain no lines) have some of the properties of sharp weakly locally compact ones, this in spite of the fact that they are more numerous. This comes from the fact that they can be embedded in a product [0, +∞[^{J} where the compactness of closed bounded intervals is a useful argument. C. Lescarret [9] used G. Choquet’s results some years ago (he used the fact that when the sum of two epigraphs is closed the infimum convolution of the functions is exact and lower semi-continuous — briefly lsc). In a forthcoming paper [12] M. Volle also uses G. Choquet’s results. We will consider, for a convex function f, the property “Epif is sharp and weakly complete” and will try to give a characterization of this property using the polar function f^{∗}. The result is not as good as J.L. Joly’s characterization of “Epif is sharp and weakly locally compact”. The Joly characterization is ([8, Ch.I §§4–5], [3, Th.I.14 pp. 15–16]) : “the polar f^{∗} is finite and continuous for the Mackey topology at least in one point.” We do not obtain an equivalence such as “Epif is sharp and weakly complete” ⟺ “f^{∗} is finite on an affinely generative convex set,” but the result is not so far from this equivalence. We thank Michel Volle for his remarks on the preliminary versions.

2. Notations

If E is a Hausdorff lctvs, E′ denotes its topological dual and E^{∗} its algebraic dual. Hence

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

102 J. Saint-Pierre, M. Valadier / Functions with sharp weakly complete epigraphs

E′ is the set of all continuous linear forms on E. There is a canonical injection of E into E′^{∗} and the weak topology σ(E′^{∗}, E′) induces on E the weak topology σ(E, E′). Moreover E′^{∗} is weakly complete and can be identified to the completion^{1} of (E, σ(E, E′)) (we will not use this result; more generally see the Grothendieck theorem about the completion of a lctvs [2, Ch. III 3 n^{∘}6 p. III.20]). We denote by Vect(A) the vector subspace of E generated by a subset A of E. Let K be a given set. Then IR^{K} denotes the classical product space of all families (x_{k})_{k∈K} of real numbers and IR^{(K)} the vector subspace of IR^{K} (which is a strict subspace only when K is an infinite set) of families (y_{k})_{k∈K} with finite supports. The spaces IR^{K} and IR^{(K)} are in canonical duality thanks to the bilinear form

((x_{k})_{k∈K}, (y_{k})_{k∈K}) ↦ ∑_{k∈K} x_{k} y_{k}

With this bilinear form, IR^{K} is a representation^{2} of the algebraic dual of IR^{(K)}. Observe that the weak topology σ(IR^{K}, IR^{(K)}) is nothing else than the product topology of IR^{K}. If F is a vector space on the field IR and (f_{k})_{k∈K} is an (algebraic) basis of F, then the dual pair (F^{∗}, F) is isomorphic to the pair (IR^{K}, IR^{(K)}). A convex set is sharp if it does not contain any line. The support function of a subset X of E is

δ^{∗}(x′ | X) = sup {⟨x′, x⟩ : x ∈ X} .

The notation Γ_{0}(E) denotes the set of convex lsc functions on E which are proper (i.e. with values in ]−∞, +∞] and not identically the constant +∞). If f is a function from E to [−∞, +∞], its domain Domf is {x ∈ E : f(x) < +∞} and its polar f^{∗} is the function on E′ defined by

f^{∗}(x′) = sup {⟨x′, x⟩ − f(x) : x ∈ E} .

3. Main result and examples

Before giving our characterization, recall that C. Lescarret [9, just before Prop.3] says “Epif is sharp and weakly complete is equivalent to : the closed affine subspace spanned by Domf^{∗} is E′ and the set of affine continuous functions less than f^{∗} is closed in IR^{E′} for the pointwise convergence topology.””

The following Lemma is an easy result, even in infinite dimension.

Lemma 3.1. Let F be a vector space and (e_{i})_{i∈I} a family in F. The following properties are equivalent :

(i) the family (e_{i})_{i∈I} is affinely free and generative in F (i.e., for any fixed i_{0}, the family (e_{i} − e_{i_{0}})_{i∈I\{i_{0}}} is a basis of F).

1 It would be better to say E′^{∗} is a realization of the completion of E. 2 For the notion of representation of a dual and the notion of isomorphism of duality pairs

see [10, pp.2–3].

(ii) the family ((e_{i}, −1))_{i∈I} is an algebraic basis of F × IR.

Theorem 3.2. Let f ∈ Γ_{0}(E). Consider the following property :

(P) there exists a family (e_{i})_{i∈I} which is affinely generative in E′ and which verifies ∀i ∈ I, f^{∗}(e_{i}) is finite. Then (P) is a necessary condition for Epif to be sharp and weakly complete. Conversely if E is weakly complete, (P) is a sufficient condition for Epif to be sharp and weakly complete.

Proof. 1) First suppose E weakly complete and (P). Replacing if necessary I by a smaller set (see for example [1, 7 n^{∘}1 Th.2 p.147]) one may suppose that the family (e_{i})_{i∈I} is affinely free and generative. By the Lemma, ((e_{i}, −1))_{i∈I} is a basis of E′ × IR. One knows that f^{∗}(e_{i}) = δ^{∗}((e_{i}, −1) | Epif). Let identify E′ × IR with IR^{(I)}, then E × IR may be identified with a subset of IR^{I}. Then Epif is sharp thanks to the inclusion

−Epif ⊂ ∏_{i∈I} [−f^{∗}(e_{i}), +∞[ .

Since E is weakly complete, E × IR and its weakly closed subset Epif are also weakly complete (see Example 2 below which shows that the completeness of E is an essential hypothesis).

2) Now we suppose Epif sharp and weakly complete. From G. Choquet’s result [4, Prop.1] (see the Comment below for the proof), there exists a basis ((e_{i}, r_{i}))_{i∈I} of E′ × IR such that ∀i, δ^{∗}((e_{i}, r_{i}) | Epif) is finite. Necessarily the r_{i} are ≤ 0. If all the r_{i} are < 0, one has still a basis replacing each (e_{i}, r_{i}) by

\frac{1}{|r_{i}|}(e_{i}, r_{i}) = (\frac{1}{|r_{i}|}e_{i}, −1)

and the support function of Epif remains finite at these points. Hence we may suppose r_{i} = −1. Then f^{∗}(e_{i}) is finite and, thanks to the Lemma, the e_{i} are affinely free and generative. Now consider the case when some r_{i} are null. They cannot be all 0 since then ((e_{i}, r_{i}))_{i∈I} would span only a strict subspace of E′ × IR. Hence there exists i_{0} verifying r_{i_{0}} < 0. Then the family :

((ε_{i}, ρ_{i}))_{i∈I} := ((e_{i}, r_{i}) + (e_{i_{0}}, r_{i_{0}}))_{i∈I}

is still a basis of E′ × IR. Indeed, for any finite subset A of I containing i_{0}, ((e_{i}, r_{i}) + (e_{i_{0}}, r_{i_{0}}))_{i∈A} is still a basis of Vect({(e_{i}, r_{i}) : i ∈ A}) (elementarily, if (b_{1}, …, b_{n}) is a basis of a vector space, (b_{1} + b_{n}, b_{2} + b_{n}, …, b_{n} + b_{n}) is also a basis of the same space). Finally ∀i, δ^{∗}((ε_{i}, ρ_{i}) | Epif) < +∞, since the support function δ^{∗}( . | Epif) is sub-additive.

Comment. We give (following Bourbaki [2, exercise 16 in Ch.II §6, p.II.91]; and for sharp cones [7, vol.2 Prop.30.10 p.197]) a proof of G. Choquet’s result in the case where the set X is sharp. So consider a sharp weakly complete convex set X in a lctvs G and let us prove that there exists a basis of G′ at each element of which the support function of X is finite. Let X^{∘} denote the polar set in G′

X^{∘} = {x′ ∈ G′ : ∀x ∈ X, ⟨x′, x⟩ ≤ 1} .

First assume 0 ∈ X. Then, since X is σ(G′^{∗}, G′) closed, its bipolar, X^{∘∘}, calculated in G′^{∗}, equals X. We prove that Vect(X^{∘}) = G′ : Indeed, otherwise there would exist x ∈ G′^{∗}\{0} such that x is null on Vect(X^{∘}). But the line IR x would be contained in X^{∘∘}. Finally there exists a basis (g_{i})_{i∈I} of G′ = Vect(X^{∘}) contained in X^{∘} (again this comes from [1, §7 n^{∘}1 Th.2 p.147]) and the support function of X verifies ∀i, δ^{∗}(g_{i} | X) ≤ 1. For a general X, consider X_{0} = X − x_{0}, where x_{0} is any point of X.

Example 3.3. (This example shows that the property “Epif is sharp and weakly complete” is strictly weaker than “Epif is sharp and weakly locally compact” and is more algebraic). Let K be an infinite set, E = IR^{K} equipped with the product topology (hence IR^{(K)} is a representation of E′), f the indicator function of [0, +∞[^{K}. Then f^{∗} is the indicator function of the cone −([0, +∞[^{K} ∩ IR^{(K)}). But this cone has no interior point for any vector space topology, hence has no interior point for the Mackey topology τ(IR^{(K)}, IR^{K}).

Example 3.4. (Showing that for the sufficiency, one cannot remove the hypothesis that E is weakly complete). Let K be an infinite set and E = IR^{(K)} equipped with the topology σ(IR^{(K)}, IR^{(K)}). Thus IR^{(K)} is a representation of E′ and IR^{K} is a representation of E′^{∗}. Let f be the indicator of

C = −([0, +∞[^{K} ∩ IR^{(K)}) .

Then f^{∗} is the indicator of the cone [0, +∞[^{K} ∩ IR^{(K)}. Condition (P) is verified taking the canonical basis of IR^{(K)} and 0. But Epif = C × [0, +∞[ is not weakly complete. This comes from the fact that C (or −C) is not closed in IR^{K} For example, (x_{k})_{k∈K}, defined by x_{k} ≡ 1, is the σ(IR^{K}, IR^{(K)}) limit of the generalized sequence (x^{A})_{A} whose indexes are the finite subsets of K, with x_{k}^{A} = 1 if k ∈ A, 0 otherwise.

Example 3.5. (Sharpness without weak completeness does not imply (P) ). Let E = ℓ^{2}(IN^{∗}) and

f(x) = ∑_{n=1}^{∞} \frac{1}{2n} x_{n}^{2} .

Clearly Epif is sharp. One can prove [3, Th.I.34 p.33] that Domf^{∗} is the following strict vector subspace of ℓ^{2}(IN^{∗}) :

Domf^{∗} = \{y ∈ IR^{IN^{∗}} : ∑_{n=1}^{∞} \frac{n}{2} y_{n}^{2} < +∞\}

J. Saint-Pierre, M. Valadier / Functions with sharp weakly complete epigraphs 105

and that, on Domf^{∗},

f^{∗}(y) = ∑_{n=1}^{∞} \frac{n}{2} y_{n}^{2} .

Since Domf^{∗} is strictly included in E′ = ℓ^{2}(IN^{∗}), (P) is not verified. The existence of a non-convergent Cauchy generalized sequence in Epif can also be proved as follows : There exists a linear form ξ ∈ [ℓ^{2}(IN^{∗})]^{∗}\{0} null on the vector subspace Domf^{∗}. So for the bipolar f^{∗∗} calculated in [ℓ^{2}(IN^{∗})]^{∗}, one has f^{∗∗}(ξ) = 0. Indeed

f^{∗∗}(ξ) := sup {⟨ξ, y⟩ − f^{∗}(y) : y ∈ Domf^{∗}} ,

is 0 since f^{∗} is ≥ 0 and f^{∗}(0) = 0. Thus (ξ, 0) is weakly adherent to Epif. Hence there exists a generalized sequence in Epif which converges weakly to (ξ, 0) in [ℓ^{2}(IN^{∗})]^{∗} × IR. This generalized sequence is Cauchy but does not converge in Epif.

References

[1] N. Bourbaki: Algèbre, Chap. 2: Algèbre linéaire, Third edition, Hermann, Paris (1962)

[2] N. Bourbaki: Espaces vectoriels topologiques, Chap. 1 à 5, Masson, Paris (1981)

[3] C. Castaing, M. Valadier: Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. Vol. 580, Springer-Verlag Berlin (1977)

[4] G. Choquet: Ensembles et cônes convexes faiblement complets, C.R. Acad. Sci. Paris Sér. A Vol. 254 (1962), 1908–1910

[5] G. Choquet: Ensembles et cônes convexes faiblement complets, C.R. Acad. Sci. Paris Sér. A, Vol. 254 (1962) 2123–2125

[6] G. Choquet: Les cônes faiblement complets dans l’analyse, Proceedings of the International Congress of Mathematicians, Stockholm (1962), Institut Mittag-Leffler 317–330 (1963)

[7] G. Choquet: Lectures on Analysis, Benjamin, New York (1969)

[8] J.L. Joly: Une famille de topologies et de convergences sur l’ensemble des fonctionnelles convexes, Thèse de Doctorat d’État, Grenoble (1970)

[9] C. Lescarret: Sur la sous-différentiabilité d’une somme de fonctionnelles convexes semicontinues inférieurement, C.R. Acad. Sci. Paris Sér. A, Vol. 262 (1966) 443–446

[10] M. Valadier: Représentation de duals ,Sém. Anal. Convexe, Vol. 16 (1986) 6.1–6.7

[11] M. Volle: Sous-différentiel d’une enveloppe supérieure de fonctions convexes, C.R. Acad. Sci. Paris Sér. I, Vol. 317 (1993) 845–849

[12] M. Volle: A general formula for the subdifferential of an upper envelope of convex functionals, to appear.