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Abstract convex optimal antiderivatives

Having studied families of antiderivatives and their envelopes in the setting of classical convex analysis, we now extend and apply these notions and results in settings of abstract convex analysis. Given partial data regarding a c-subdifferential, we consider the set of all c-convex c-antiderivatives that comply with the given data. Under a certain assumption, this set is not empty and contains both its lower and upper envelopes. We represent these optimal antiderivatives by explicit formulae. Some well known functions are, in fact, optimal c-convex c-antiderivatives. In one application, we point out a natural minimality property of the Fitzpatrick function of a c-monotone mapping, namely that it is a minimal antiderivative. In another application, in metric spaces, a constrained Lipschitz extension problem fits naturally the convexity notions we discuss here. It turns out that the optimal Lipschitz extensions are precisely the optimal antiderivatives. This approach yields explicit formulae for these extensions, the most particular case of which recovers the well known extensions due to McShane and Whitney.


Introduction
Abstract convex analysis (or generalized convexity) has attracted more and more attention in recent years. Now this topic is studied continually, both theoretically and from the point of view of applications. It has turned out to be a useful tool for solving optimization problems which The following well known definitions are the foundations of our discussion. Unless otherwise specified, throughout the paper X and Y are arbitrary sets and c : X × Y → R is an arbitrary function. We say that a function f : X → (−∞, +∞] is proper if dom(f ) := {x ∈ X | f (x) < ∞} is not empty. The c-transform of a function is also known as its c-conjugate function. This generalization of Fenchel's conjugate function from classical convex analysis was introduced and studied by Moreau in [11]. In other, more general, discussions, one encounters convexity of a function with respect to a family of elementary (sometimes referred to as affine) functions defined on X. That is, a function is said to be convex with respect to a set of functions A if it is the upper envelope of a subset of A. In our discussion, A = {c(·, y) + r | y ∈ Y, r ∈ R}. Sometimes, the function c is allowed to take the values ±∞. Many other distinctions within our definitions and beyond them, and the theories to which they lead, were considered by different authors; see, for example, [16,17]. In our discussion we focus our attention on the settings in the above definitions. When referring to the theoretical study of c-convexity, the function c is called a coupling function between X and Y , while in the particular application to the study of optimal transport, as in [18], it is said to be the cost function.

Clearly,
c(x, y) ≤ f (x) + f c (y) for all x ∈ X and y ∈ Y.
The case of equality is captured in the following definition of the c-subdifferential. We will denote the graph of a multivalued mapping M : When ∂ c f (x) = ∅, we say that f is c-subdifferentiable at x. When M : X ⇒ Y and G(M ) ⊂ G(∂ c f ), we say that f is a c-antiderivative of M .
For a proper function f : X → (−∞, +∞], it is clear that dom(∂ c f ) ⊂ dom(f ), that is, f is finite where it is c-subdifferentiable. We will continue our discussion of elementary c-convex analysis in Section 2.
Our main interest here is in solving the following problem and then applying our solution in particular situations. Suppose that f : X → (−∞, +∞] is a c-antiderivative of the mapping M : X ⇒ Y and suppose that S is a nonempty subset of dom(M ). We are interested in the c-convex solutions h : X → (−∞, +∞] of the problem One may consider this as a constrained c-convex extension problem. We are interested in extending the function f | S to a c-convex function defined on X while keeping it a c-antiderivative of M . In It is well known that Rockafellar's famous characterization [13] of cyclically monotone mappings from classical convex analysis extends to c-cyclically monotone mappings with respect to c-convexity. Moreover, Rockafellar's antiderivative plays the same rôle as in the classical case. It is an explicitly represented c-convex c-antiderivative of a given c-cyclically monotone mapping. In Section 4 we recall these facts and then apply Rockafellar's antiderivative in order to explicitly construct the minimal antiderivative in the general case of (6). Besides this advantage, this approach is an alternative way to the one in Section 3. It enables us to solve problem (6) without appealing to duality results.
Let (X, d) be a metric space. Recall that a function f : for all x, y ∈ X.
Let S be a nonempty subset of X and let f : S → R be K-Lipschitz. Then the well known extension theorem of McShane [10] and Whitney [19] asserts that f extends to a K-Lipschitz function which is defined on all of X. In particular, the minimal and maximal K-Lipschitz extensions of f are given explicitly by respectively. In Section 5 we address the following constrained Lipschitz extension problem: Suppose that f : A ⊂ X → R is a K-Lipschitz function and let G be a subset of A × A such that We set G = G(M ) where M : A ⇒ A is the mapping the graph of which is the set G. Given a nonempty subset S of dom(M ), the problem is to extend f | S to a K-Lipschitz function which is defined on all of X and still satisfies (8). In particular, we look for the minimal and maximal extensions. We accomplish this and provide formulae for the minimal and maximal K-Lipschitz extensions that satisfy (8). Apart from particular cases, formulae (7) are not necessarily solutions to this generalized problem. If one assumes the McShane and Whitney extension result, then the existence of optimal Lipschitz extensions in our discussion is a rather simple consequence. However, approaching this problem through −d-convexity, we do not need to rely on the McShane and Whitney theorem. Furthermore, this approach leads to the representation of the optimal extensions with explicit formulae of which formulae (7) are a particular case. When c = −d, then −d-convex functions are precisely the 1-Lipschitz functions. In this case, f is a −d-antiderivative of M and the problem fits naturally in the context of optimal antiderivatives. We note that this discussion is never empty since the identity mapping can play the rôle of M with respect to any function f . This particular case recovers the McShane and Whitney extensions from our discussion. Several approaches to the theory of Lipschitz functions and to the problem of extending Lipschitz functions with tools from classical convex analysis and abstract convex analysis have been proposed in the last three decades. For instance, in Section 5 we generalize previous dicussions and results in this direction which appear in [6,8,9,15,17].
In the framework of classical convex analysis and monotonicity, the associated Fitzpatrick function of a monotone mapping was first defined and studied in [7]. Fitzpatrick also considered an associated family of functions with a similar underlying connection to the given monotone mapping, now known as the Fitzpatrick family. It was the beginning of what has become an intense study, often referred to as the "representation of monotone operators by convex functions". Apart from their theoretical interest, these functions found application in new accessible convex analytic proofs of sum and range theorems for monotone mappings and of other results in monotone operator theory. A well known property of the Fitzpatrick function, which was originally observed by Fitzpatrick in [7], is that when the underlying mapping is maximal monotone, then the Fitzpatrick function is the minimal function in the Fitzpatrick family. In [2] we studied how a monotone mapping gives rise to an associated family of convex antiderivatives and we identified the Fitzpatrick function as the minimal convex antiderivative. This extended the notion of minimality of the Fitzpatrick function to also hold for monotone mappings which are not necessarily maximal monotone. In Section 6 we generalize most of the discussion in [2] and show that it continues to hold in the setting of c-convexity and c-monotone mappings.

c-Convex Analysis Preliminaries
We proceed with our discussion of abstract convexity by recalling some basic properties of c-convex functions and c-antiderivatives, which we utilize later and which are easily verified using the definitions: Recall that the indicator function of a set S ⊂ X is the function ι S : X → (−∞, +∞], defined by • For every y ∈ Y , the function c(·, y) : X → (−∞, +∞] is c-convex since c(·, y) = ι c {y} .
Proof . Suppose that y ∈ ∂ c f (x). Then for every x ′ ∈ X we have which implies that y ∈ ∂ c g(x).
It is a crucial and elementary fact in our discussion that Definitions 1.2 and 1.3 are preserved by upper envelopes. Unless otherwise specified, throughout the paper the families we consider are indexed by an arbitrary index set. Proof . Suppose that for every s, the function f s is c-convex, say f s = g c s . Then the function g := inf s g s does not attain the value −∞, that is g : Y → (−∞, +∞]. Indeed, suppose that for some y 0 ∈ Y we have inf s g s (y 0 ) = −∞. Let {s n } be a sequence such that g sn (y 0 ) → −∞. Then for every x ∈ X, we obtain which implies that f cannot be a proper function in this case. A straightforward verification now yields f = g c . Proof . Let (x, y) ∈ M and x ′ ∈ X. Since f s is a c-antiderivative of M for every s, we have for every s. It follows that f (x) + c(x ′ , y) − c(x, y) ≤ f (x ′ ), and consequently, that f is a cantiderivative of M .
A major part of the theory of c-convex functions is made possible by the following well known fundamental property of the c-transform [11]: Proof . By definition, for any y ∈ Y we have Letting y ′ = y, we see that f ccc (y) ≤ f c (y), while letting x ′ = x we obtain that f ccc (y) ≥ f c (y). Now, if f is c-convex, then f = g c for some function g : Y → (−∞, +∞] and so f cc = g ccc = g c = f . Conversely, if f = f cc , then f is c-convex by definition because it is the c-transform of the function f c : Y → (−∞, +∞].
We can now justify the use of the notion of c-convexification as follows: Proof . By definition, for any x ∈ X we have Letting is a c-convex function such that h ≤ f , then applying the fact that the c-transform reverses order and the above criterion for c-convexity, we get h = h cc ≤ f cc ≤ f . Furthermore, we can now relate the c-sundifferential of a function with the c-subdifferential of its c-transform. Recall that the mapping Proof . Let f be a c-antiderivative of M . If f is c-convex, then f cc = f . It follows that the set of points (x, y) where the equality c(x, y) = f cc (x) + f c (y) holds is precisely the set of points where the equality c(x, y) = f (x) + f c (y) holds. This implies that ∂ c f c = (∂ c f ) −1 . In the general case, since f cc ≤ f , then for every (x, y), It now follows that the set of points (x, y) where the equality c(x, y) = f (x) + f c (y) holds is contained in the set of points where the equality c(x, y) = f cc (x) + f c (y) holds, that is, Finally, if f is c-subdifferentiable at x, say y ∈ ∂ c f (x), then both the inequalities in (13) become equalities and we obtain f At this point we describe the manner in which classical convex analysis is captured by c-convex analysis. Let X be a locally convex separated topological vector space, let X * be its dual and set x * , x = x * (x) for x ∈ X and x * ∈ X * . Given a function f : X → (−∞, +∞], its Fenchel conjugate is the function f * : X * → (−∞, +∞] defined by We see that when we let Y = X * and c(x, y) = y, x for x ∈ X and y ∈ Y , then f c = f * for a function f : X → (−∞, +∞]. However, we have f cc = f * * | X , when we identify X with its canonical embedding in X * * . In this setting, if f is proper, then it is well known that f is convex and lower semicontinuous if and only if f = f cc . The characterization of c-convex functions obtained in Proposition 2.4, namely, that the proper function f satisfies f = f cc if and only if f is c-convex, is not a direct generalization of the above fact from classical convex analysis. It is, of course, true that in the classical setting, the proper function f is c-convex if and only if f = f cc . However, the fact that f is c-convex if and only if f is lower semicontinuous and convex is not a consequence of Proposition 2.4. We are thus led to consider the following observations. Recall that the epigraph of a function f : Perhaps, since a proper function is c-convex if and only if its epigraph is the non-empty intersection of a family of epigraphs of c-convex functions, one should say that a "c-convex" function is, in fact, "outerly c-convex". In the classical definition of a convex function, convexity can be referred to as "inner convexity", that is, through line segments inside the epigraph. For a proper and lower semicontinuous function, what unifies "inner convexity" and "outer convexity" in the classical case, that is, the reason that a proper function is "innerly c-convex" (convex and lower semicontinuous) if and only if it is "outerly c-convex" (c-convex), is the Hahn-Banach separation theorem.
In Sections 5 and 6 we study families of functions which turn out to be families of c-convex c-antiderivatives. The families in Section 5 are convex. The fact that the families in Section 6 are closed under nontrivial convex combinations is well known in the classical case. Before ending this section we draw the reader's attention to the c-convexity and c-subdifferentiability properties of nontirivial convex combinations of functions. We refer to these properties in the next section where we define families of c-convex c-antiderivatives. Proof . Let (x, y) ∈ G(M ). Then y ∈ ∂ c g(x) and y ∈ ∂ c h(x). For every x ′ ∈ X, summing up the inequalities and Convex combinations of c-convex functions may fail, in general, to be c-convex. Examples are easily constructed. In particular, we end this section with the following example where it is also easy to identify all c-convex functions on X and Y .
Then the functions in Γ c (X), that is, the c-convex functions on X, are the functions of the following types: Consequently, Γ c (X) is not closed under nontrivial convex combinations. In particular, the functions x and |x| are c-convex c-antiderivatives of the mapping M : On the other hand, all proper functions on Y are c-convex.
Proof . We prove (14). Since there are 3 types of proper functions g : Y → (−∞, +∞], these functions have 3 types of c-transforms, that is, 3 types of c-convex functions on X: In cases 1 and 2 we see that all real values for p and q in (14) are possible. In the third case, since the transformation (g(a), g(b)) → (1/2)(g(a) − g(b), g(a) + g(b)) is onto R 2 , all real values for r and s in (14) are possible. In all three cases, c-transforming the function g c , we get g = g cc .

The Family A [c,f | s ,M ] of Antiderivatives and Duality
Our main purpose is to extend the notions and results from classical convex analysis which we have presented in [2] to the setting of c-convexity. To this end, we first consider families of c-convex c-antiderivatives as follows.
In the above setting, since M (S) is a subset of dom(M −1 ), Corollary 2.6 makes it is also possible to consider the c-dual problem: f c is a c-antiderivative of M −1 . We therefore denote the set of c-convex solutions h : Y → (−∞, +∞] of the problem In the following result we use the c-convexification in order to establish the nonemptiness of Then the existence of a maximal element follows. We collect duality relations between which, in particular, imply the existence of a minimal member. By c-transforming the maximal element of one family we arrive at the minimal element of the dual family and vice versa.
as well as its lower envelope, the function Furthermore, In the case where S = dom(M ), we have and In this case, if h : Proof . In order to obtain the nonemptines of A [c,f | S ,M ] , we employ Corollary 2.6. Since f is a proper c-antiderivative of M , the function f cc is a c-antiderivative of M . Since S is a set of points where f is c-subdifferentiable, f cc agrees with f at any point in S. The c-convexity of f cc now implies that Applying the upper envelope properties 2.2 and 2.3, we conclude that the upper envelope satisfy (17). Then, again, according to As a consequence of this discussion and of (18), we conclude, in particular, that for any function h : We now consider the case where S = dom(T ). Since f is a c-antiderivative of M , we may apply Proposition 2.1 and conclude that the function f + ι dom(M ) is the greatest c-antiderivative M that agrees with f on dom(M ). It follows that its c-convexification

Note that an arbitrary function in
, which completes the proof of (20). In order to prove (21) we proceed in a similar way. Since f c is a c-antiderivative of M −1 , applying Proposition 2.1, we conclude that the function f c + ι Im(M ) is the greatest antiderivative of M −1 that agrees with f c on Im(M ).
Since h was arbitrary, we conclude that (f c + ι Im(M ) ) c is the lower envelope of A [c,f | dom(M ) ,M ] . This verifies the left equality of (21). The right equality of (21) is evident from the following computation: Finally, in order to prove (22), we first note that the implication ⇒ is trivial. We see that the consequences of Proposition 2.4 play a crucial role in the proof. In particular, the nonemptiness of A [c,f |s,M ] and the existence of a minimal member follow from these duality results. A different approach, which does not employ these duality properties of the c-transform and which gives rise, constructively, to a minimal member, is presented in the next section. In the particular case where S = dom(M ), such an argument is already at hand.
It follows that α| dom(M ) = f | dom(M ) . Consequently, for every (s, t) ∈ M and x ∈ X, we see that which implies that G(M ) ⊂ G(∂ c α). Since α is the proper upper envelope of c-convex functions and since it satisfies (17), we conclude that α ∈ A [c,f | dom(M ) ,M ] . If h is any function which satisfies (17), then for every x ∈ X, We remark in passing that (18) is an extension of (16) from [2], which dealt with the classical case. Clearly, the converse implication there only holds when h is lower semicontinuous and convex.
Before ending this section we wish to make a remark regarding the closedness of A Observe that in this example any c-convex function h such that h| S = f | S is automatically a c- We have already seen that any nontrivial convex combination of x and |x| is not c-convex. Hence, A [c,f | S ,M ] is not closed under nontrivial convex combinations in this case.
Finally, recalling Proposition 2.7, we note the following Corollary.

Cyclic Monotonicity and Minimal Antiderivatives
We begin this section by recalling the following notions.
In this case we say that G(M ) is an n-c-monotone set. A mapping M is said to be cyclically monotone with respect to c, c-cyclically monotone for short, if it is n-c-monotone for all n ∈ N. A 2-c-monotone mapping is simply called a c-monotone mapping. Explicitly, the mapping M is c-monotone if for all (x 1 , y 1 ), (x 2 , y 2 ) ∈ G(M ), we have The mapping M is said to be maximal n-c-cyclically monotone if G(M ) has no proper n-c-cyclically monotone extension in X × Y .
We proceed by recalling Rockafellar's characterization of cyclically monotone mappings. This characterization also holds in the setting of c-monotonicity and so does its proof.  Proof . Suppose first that f : Summing up these inequalities over i, we see that M is c-cyclically monotone. Conversely, suppose that M is proper and c-cyclically monotone and let s ∈ dom(M ). For every For each m ∈ N and 1 ≤ j ≤ mk + 1, we define (u j , v j ) = (x i , y i ) when j mod k = i. Let t ∈ M (s). For every x ∈ X, we have Proof . We define α : X → (−∞, +∞] by x ∈ X.
Indeed, let x ∈ X. For n ∈ N, which implies that R [c,M,s] (x) ≤ h(x), as claimed.

Optimal Antiderivatives as Constrained Optimal Lipschitz Extensions
In our discussion of Lipschitz functions we will assume that the functions are 1-Lipschitz. Since Kd α , where 0 < α ≤ 1 and K > 0, is also a metric whenever d is, our results and formulae are also easily extensible to α-Hölder continuous functions with constant K by replacing d with Kd α . First we discuss the Lipschitz extension problem without introducing abstract convexity into the discussion. We begin with the following envelope properties of sets of Lipschitz functions. We omit the direct proof. We will offer a proof in the setting of −d-convexity shortly. We note in passing that the particular case will follow from the general case when we let M = I, where I : X → X is the identity on X. Next, we recall the following well-known extension theorem of McShane [10] and Whitney [19].
Theorem 5.2 Let S be a nonempy subset of a metric space (X, d) and let f : S → R be 1-Lipschitz.
Then f extends to a 1-Lipschitz function which is defined on all of X. In particular, the functions Proof . If h is a 1-Lipschitz extension of f , then for x ∈ X and every s ∈ S, we have It follows that α ≤ h ≤ γ. Since f is 1-Lipschitz on S, we have α| S = γ| S = f | S . In particular, α and γ are proper. Now Proposition 5.1 guarantees that α and γ are 1-Lipschitz.
These facts suffice to establish the existence of optimal extensions for the following problem. Proof . Since f extends to a 1-Lipschitz function which is defined on X, this extension clearly extends f | S and satisfies (30). Thus, the set of 1-Lipschitz extensions of f | S which satisfy (30) is not empty and has proper lower and upper envelopes. From Proposition 5.1 it follows that these envelopes belong to the set. Now we introduce abstract convexity into our discussion in order to recover all the above facts from a different perspective. This will allow us to construct the optimal extensions explicitly. We do not assume the McShane and Whitney extension theorem; it follows as a particular case. When (X, d) is a metric space, Y = X and we take the function c : X × Y → R to be c = −d, the following relations between c-convex functions and 1-Lipschitz functions emerge.
Proof . If f is 1-Lipschitz, then for every y ∈ X we have f −d (y) = sup x∈X −d(x, y)−f (x) ≤ −f (y). Since equality holds for x = y, we have verified that 1 ⇒ 2. If f −d = −f , then for every x ∈ X, which implies that f = (f −d ) −d and consequently that f is −d-convex. This verifies 2 ⇒ 3. For every y ∈ X, the function −d(·, y) is a −d-antiderivative of the identity I. Indeed, let x ∈ X. For any y))(x). Since this holds for every x ∈ X, we obtain G(I) ⊂ G(∂ −d (−d(·, y))). It follows that −d(·, y) + const is also a −d-antiderivative of the identity I for any constant. Now suppose that f is −d-convex, say f = g −d = sup y∈X −d(·, y) + g(y). Then f is the proper upper envelope of a family of −d-antiderivatives of I. It now follows from Proposition 2.3 that f is also a −d-antiderivative of I, which concludes the proof of 3 ⇒ 4. If f is a −dantiderivative of I, then for every x, x ′ ∈ X, we have x ∈ ∂ −d f (x) and consequently we obtain Remark 5.5 Since we have not used the fact that d(x, y) = 0 only if x = y, Proposition 5.4 also holds for a pseudometric d. The fact that the identity mapping is −d-cyclically monotone is an immediate consequence of the properties of the metric d.
It follows that the identity mapping is the most trivial −d-subdifferential of −d-convex functions. It is another example of a situation which is impossible in classical convex analysis: the c-subdifferential of a c-convex function need not be maximal c-cyclically monotone or maximal c-monotone. Furthermore, it does not determine its c-antiderivative up to an additive constant. In the classical setting, where proper, convex and lower semicontinuous functions are precisely the c-convex functions, when we take c to be the pairing between a Banach space and its dual, these are well known facts due to Rockafellar [13,14]. That is, the subdifferential of a proper, convex and lower semicontinuous function determines its antiderivative up to an additive constant. It is maximal monotone and maximal cyclically monotone. This is illustrated by the following example which extends Example 2.3 from [15].
Example 5.6 The identity mapping I on a metric space is −d-cyclically monotone. G(I) is contained in the graph of the −d-subdifferential of any −d-convex function. As a consequence, any −dcyclically monotone set G(M ) ⊂ X ×X extends to the −d-cyclically monotone set G(M )∪G(I). In particular, G(I) is extensible by one point to a −d-cyclically monotone set, using any point in X×X. Therefore, if X is not a singleton, I is not maximal −d-cyclically monotone and it is not maximal −d-monotone. However, it is the (whole) −d-subdifferential of any function f : X → R such that f (y) − f (x) < d(x, y) for all x = y. In particular, it is the −d-subdifferential of any K-Lipschitz function, where 0 ≤ K < 1. Therefore, I does not determine its −d-convex −d-antiderivative up to an additive constant.
Proof . The fact that I is −d-cyclically monotone and the fact that G(I) is a subset of G(∂ −d f ) for any −d-covex function f are clear from Proposition 5.4. If G(M ) is a −d-cyclically monotone set, then M admits a −d-convex −d-antiderivative f . It follows that G(M ) ∪ G(I) ⊂ G(∂ −d f ), which is cyclically monotone. Since a singleton in X × X is always −d-cyclically monotone, G(I) extends to a −d-cyclically monotone set by unifying it with any singleton. If f : d(x, y) for all x = y, then f is −d-convex. In this case, is the minimal 1-Lipschitz function that agrees with f on S and satisfies (32). The function is the maximal 1-Lipschitz function that agrees with f on S and satisfies (32). and In particular, suppose that f : S ⊂ X → X Proof . We let f (x ′ ) = ∞ for x ′ ∈ X \ dom(M ). Now it follows that for every (x, y) ∈ G(M ) and every In the first equality we used the fact that f −d = −f . In (39) we used the fact that for each In order to obtain (34), we apply (19) as follows Together with (39), this implies implies (34). A remark regarding the history of some parts of our discussion in this section is in order. Some of the previously published results regarding the subject matter of this section were stated in settings where X = R n and some in the generality of metric spaces. Some have been stated for Lipschitz functions and some for Hölder continuous functions. As we remarked at the beginning of this section, we do not distinguish between these here. The fact that the −d-convex functions are precisely the 1-Lipschitz functions had already appeared in [6]. It has evolved [9] into a pointwise variant of the characterization 1 ⇔ 2 ⇔ 3 ⇔ 4 of Proposition 5.4. This evolvement was recalled later in [17]. Recently, this has been an example of a coupling function c and the corresponding set of c-convex functions which is given by different authors; see, for example, [15,18]. Regarding the extension of functions, given a function f : S ⊂ X → R, we set f (x) = ∞ for x ∈ X \ S. In [8] the McShane and Whitney theorem was reestablished with the aid of the infimal convolution from classical convex analysis in R n . Given x ∈ R n , the value of the maximal extension of f at x is then the infimal convolution of f with the norm. In [9], c-conjugation theory was applied with the function c = −d. Then, the maximal extension of McShane and Whitney was recognized as (f −d ) −d = −f −d , while the minimal extension was recognized as (−f ) −d . In [9] the study continued to the case where the Lipschitz constant is allowed to vary. For a fixed α, this is done by considering the coupling function c : X ×(X ×R) → R defined by c(x, (y, K)) = −Kd α (x, y). Then, the c-convex functions on X are characterized as the functions on X which satisfy the α-growth condition. This is also recalled later in [17]. The elementary c-convex functions in our discussion, that is, the functions −Kd(·, y) + r where x ∈ X and K, r ∈ R are sometimes called cone functions. Recently, attention has been drawn to operations, like the infimal convolution in R n , which are carried out with cone functions in order to solve the problem of absolutely minimizing Lipschitz extensions; see, for example, [1].

Optimal Antiderivatives in the Representation of c-Monotone
Mappings by C-Convex Functions In [12] the discussion regarding the representation of monotone mappings by convex functions is extended to the settings of c-convexity and c-monotonicity. In particular, the definition of the Fitzpatrick function and some of its basic properties are extended to hold for c-monotone mappings (see (24) in Definition 4.1). This discussion requires a type of the following convention: given It follows, after permuting coordinates, that F C [c,T ] is the C-convexification of c + ι G(T ) . When T is c-monotone, then for any (x, y) ∈ X × Y we have It now follows that when T is maximal c-monotone, we have and each one of the inequalities becomes an equality for (x, y) ∈ G(T ).

The Fitzpatrick family of functions is defined by
It is a well-known fact in the classical case that when T is maximal monotone, then F | [ ·,· ,T ] is the minimal function in F| [ ·,· ,T ] . We make a remark regarding this case towards the end of this section. For a general function c, we conclude (as in [12]) that when T is maximal monotone, then F [c,T ] ∈ F [c,T ] and, after permuting coordinates, F C [c,T ] ∈ F [c,T ] . However, so far no minimality property of F [c,T ] is available in the generality of c-monotone mappings. The framework of optimal antiderivatives does allow us to obtain a minimality property: F [c,T ] is minimal in a certain family of C-convex C-antiderivatives which is determined by T . This is carried out below. In order to bring antiderivatives into the discussion we make use of the following notation.
On the other hand, if ∆ T is C-cyclically monotone, then, in particular, ∆ T is C-monotone and therefore T is c-monotone. Consequently, G(∆ T ) ⊂ G(∂ C (c + ι G(T ) )). This implies that 3 ⇔ 4 and completes the proof of (A).
(B) Suppose that T is c-monotone. In the setting of Theorem 3.2, we let M = ∆ T and f = c+ι G(T ) , so that G(M ) ⊂ G(∂ C f ) according to part (A). Letting S = G(T ) = dom(∆ T ), we may now consider the family A [C,c| G(T ) ,∆ T ] . Since S = dom(M ), we may apply (21) in order to compute the minimal C-antiderivative of ∆ T that agrees with c on G(T ): In the classical case, if T is monotone, then which, together with (51), implies that when T is maximal monotone, then Indeed, when X is a topological vector space, Y = X * and c = ·, · , then we can rely on the fact that if g is Gâteaux differentiable at x and g ≤ f , where f is convex and g(x) = f (x), then ∇g(x) ∈ ∂f (x). In particular, for f ∈ F [c,T ] , since ∇c(x, y) = (y, x), then f is a convex antiderivative of ∆ T , hence, f ∈ A [C,c| G(T ) ,∆ T ] . However, even in the classical case, without the notion of minimal antiderivatives, when T is monotone but not maximal monotone, there was a problem discussing the minimality of F [c,T ] . In this case, it may happen that c F [c,T ] , that is, F [c,T ] is not a member of F [c,T ] . This gap has been filled in [2], where we consider the extended family A [C,c| G(T ) ,∆ T ] . There, for a monotone mapping T , we have seen that A [C,c| G(T ) ,∆ T ] ⊃ F [c,T ] and that F [c,T ] is the minimal function in A [C,c| G(T ) ,∆ T ] . A more detailed discussion in the classical case is available in [2].
In the classical setting in [2], the equivalence 1 ⇔ 3 also followed from a direct computation, without having an antiderivative at hand. This computation extends to c-monotone mappings without difficulty. It is a cyclic generalization of the identity (52). Indeed, let {(x 1 , y 1 ), . . . , (x n , y n )} ⊂ X × Y and set (x 1 , y 1 ) = (x n+1 , y n+1 ). We compute the cyclic sum of the ordered pairs ((x i , y i )(y i , x i )), 1 ≤ i ≤ n: Letting n = 2, we get (52). We see that if the ordered pairs (x i , y i ), 1 ≤ i ≤ n, are in c-monotone relations, then the ordered pairs ((x i , y i )(y i , x i )), 1 ≤ i ≤ n, are in C-cyclically monotone relations. Conversely, if the latter are in C-cyclically monotone relations, then, in particular, they are in C-monotone relations. We now get the monotone relations of the ordered pairs (x i , y i ), 1 ≤ i ≤ n, from the case n = 2. Proof . The fact that I X is −d-monotone was discussed in Section 5. The formula for the Fitzpatrick function in (54) follows from the triangle inequality and the fact that d(x, x) = 0. Since