On Order Preserving and Order Reversing Mappings Deﬁned on Cones of Convex Functions MANUSCRIPT

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A further question we concern most is about the behavior of the “Artstein-Avidan-Milman” theorem if we substitute a general Banach space X for Rn. The question is apparently natural and worth considering, which can be divided into the following three more concrete questions. Problem 1.1. For what inﬁnite dimensional Banach spaces X, does there exist a fully order reversing mapping T : conv(X) → conv(X)? Problem 1.2. Does the “Artstein-Avidan-Milman” theorem hold again for all fully order reversing mappings T : conv(X) → conv(X)? Problem 1.3. Does the “Artstein-Avidan-Milman” theorem hold true for every order preserving mapping S : conv(X) → conv(X)?

Recall that a convex function f defined on a Banach space X is said to be proper if it is nowhere −∞-valued and with its essential domain dom(f) 6= ∅. For a partially ordered set P , a mapping T : P → P is said to be fully order preserving (resp. reversing) provided it is a bijection and satisfies f ≥ g ⇐⇒ Tf ≥ Tg ( resp. f ≥ g ⇐⇒ Tf ≤ Tg). Note that in a Banach space X the Legendre transform L : conv(Rn) → conv(Rn) becomes into the following Fenchel transform: ∗ F : conv(X) → conv∗(X ), which is defined for f ∈ conv(X) by (Ff)(x∗) = sup{hx∗, xi− f(x): x ∈ X}, x∗ ∈ X∗,

∗ ∗ where conv(X) (resp. conv∗(X )) is the cone of all extended real-valued lower semicontinuous (resp. w -lower semicontinuous) proper convex functions on X (resp. X∗). In 2015, Iusem, Reem and Svaiter [11] generalized the “Artstein-Avidan-Milman” theorem to a general Banach space X in the following manner. Theorem 1.2 (Iusem-Reem-Svaiter). Suppose that X is a Banach space. Then,

∗ i) for every fully order reversing mapping S : conv(X) → conv∗(X ) there exists an isomorphism U : X → ∗ ∗ X, x0 ∈ X, x0 ∈ X , α> 0, and r0 ∈ R so that ∗ ∗ ∗ ∗ ∗ ∗ ∗ (Sf)(x )= α(Ff)(U x + x0)+ hx , x0i + r0, f ∈ conv(X), x ∈ X ;

ii) for every fully order preserving mapping T : conv(X) → conv(X) there exists an isomorphism U : X → X, ∗ ∗ x0 ∈ X, x0 ∈ X , α> 0, and r0 ∈ R so that ∗ (Tf)(x)= αf(Ux + x0)+ hx0, xi + r0, f ∈ conv(X), x ∈ X.

The second conclusion of the “Iusem-Reem-Svaiter” theorem Theorem 1.2 ii) can be regarded as a positive answer to Problem 1.3, i.e. a perfect extension of the “Artstein-Avidan-Milman ” theorem for fully order preserving mappings. But Problems 1.1 and 1.2 remain open. The first conclusion Theorem 1.2 i) can be understood as a characterization of fully order reversing mappings T : conv(X) → conv(X∗) whenever X is reflexive. “However, the issue of characterizing fully order reversing mappings from conv(X)to itself (or, from conv(X) to conv(X∗) in nonreflexive case) has left as an open problem, deserving future research [11, p.89].” Iusem, Reem and Svaiter [11] further pointed out: In fact, we don’t even know whether there is a fully order reversing mapping T : conv(X) → conv(X) (or, conv(X) → conv(X∗) in the nonreflexive case). Order preserving isomorphisms defined on sets with specific structures had been studied for some time, in some cases in connection with applications to physics, however, Artstein-Avidan and Milman’s work [5] was the first to deal with this issue in connection with the cone of convex functions and results in [5] were the starting point of several interesting developments. For further information in this direction we refer to [2–4, 6, 11] and references therein. The main results of this paper are as follows. (Their proofs are starting from Section 7.) The next result (Theorem 1.3) presents a characterization of a Banach space X for which there exists a fully order reversing mapping on conv(X). Therefore, it gives Problem 1.1 a complete answer. Theorem 1.3. For a Banach space X there is a fully order reversing mapping S : conv(X) → conv(X) if and only if X is reflexive and linearly isomorphic to its dual X∗.

The next theorem gives Problem 1.3 an afﬁrmative answer, and it can be understood as a perfect extension of the “Artstein-Avidan-Milman” theorem for order reversing mappings.

2 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

Theorem 1.4. Suppose that X is a Banach space. Then for every fully order reversing mapping S : conv(X) → ∗ ∗ ∗ conv(X) there exists a linear isomorphism U : X → X , x0, ϕ0 ∈ X , α> 0 and r0 ∈ R so that ∗ (Sf)(x)= α(Ff)(Ux + x0)+ hϕ0, xi + r0, ∀x ∈ X, where F : conv(X) → conv(X∗) is the Fenchel transform.

Let subl(X), mink(X) and semn(X) be the three common cones of extended-real-valued lower semi continuous convex functions deﬁned on the Banach space X, successively: all sublinear functions, all Minkowski functionals and all seminorms. Theorem 1.5. Let T : subl(X) → subl(X) be a fully order preserving mapping, then there exists a linear isomor- ∗ ∗ phism E :→ X and a bounded linear functional x0 ∈ X so that ∗ (Tf)(x)= f(Ex)+ hx0, xi ∀x ∈ X, f ∈ subl(X). Theorem 1.6. Suppose that C is one of the two cones {mink(X), semn(X)}. Then for every fully order preserving mapping T : C → C there is a linear isomorphism E : X → X so that (Tf)(x)= f(Ex), ∀x ∈ X, f ∈ C.

We should also mention that our proofs are different from those of previous known results: 1. Through introducing the concepts of bounded sup-complete cone and sup-generating class of cones consisting of convex functions, and discussing their properties, we show that every fully order preserving mapping defined on a bounded sup-complete cone of convex functions is bounded-continuous with respect to the topology induced by the pointwise convergence; and is affine, whenever it is restricted to its perfect sup-generating class. These facts make the proofs of Theorems 1.3 and 1.4 much simpler. 2. Though fully order preserving mappings T : C → C, when C is subl(X), mink(X) or semn(X) (Theorem 1.5 and Theorem 1.6), share quite similar form of representations, however, their proofs are entirely different. In the case that C = subl(X) we convert sublinear functions to the set-operation of the images of their subdifferentials. To show the theorem for C = mink(X), we first extend the fundamental theorem of (finite dimensional) affine (or, projective) geometry to infinite dimensional spaces. Then, making use of this generalization, we show that every fully order preserving self-mapping defined on mink(X) is again a fully order preserving self-mapping restricted to semn(X). 3. We identify aff(X) (the space of all continuous affine functionals on X) with X∗ ⊕ R, and use the “new” property that every convex function defined on a Banach space X is a sublinear function on X ⊕ R restricted to the hyperplane X ⊕{1} ( Proposition 2.1), this fact is used to prove Theorems 7.2, 7.3 and 7.5. Our paper is organized as follows. The next section (Section 2) collects some preliminary facts from convex analysis which will be used throughout this paper. In Section 3, some notions such as sup-generating class, purity, atomic of sup-generating class will be introduced and discussed. In Section 4, some basic properties such as continuity of fully order preserving mappings defined on cones of convex functions will be discussed. We will see in Section 5 that fully order preserving mappings behave nicely when restricted to their certain sup-generating classes. In Section 6, we will return to discuss fully order preserving mappings defined on the cones conv(X) and subl(X). With the previous preparations, we are ready to show the extensions of the “Artstein-Avidan-Milman” theorem for order preserving and order reversing mappings in Section 7. In Section 8, we are devoted to the extension of fundamental theorem of affine geometry to infinite dimensional case. With the help of the extended fundamental theorem of affine geometry, the representation theorem of fully order preserving mappings defined on the cone semn(X) of seminorms is shown in Section 9. The representation theorem of fully order preserving mappings defined on the cone mink(X) of Minkowski functionals is in Section 10. In the same sprit of Section 9 and Section 10, some characterizations of fully order preserving mappings defined on sublinear functions will be provided in Section 11.

2 Convex functions and their subdifferentials

The deﬁnition and basic properties of subdifferential operator [12] will be recalled and concepts concerning with order on cones of convex functions will be introduced. We conclude this section by providing characterizations of minimum upper bound of convex functions. The letter X will always be a real Banach space and X∗ its dual. Recall that an extended real-valued convex function f : X → R ∪ {±∞} is said to be proper, if f(x) > −∞ everywhere with dom(f) := {x ∈ X : f(x) < ∞}= 6 ∅. By a cone C of convex functions, we mean that it is convex and closed under multiplication of non-negative numbers. We use conv(X) to denote the cone of all proper convex lower semicontinuous (l.s.c., for short) functions on X; aﬀ(X),

3 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

the space of all continuousaffine functions on X, i.e. aff(X)= {ϕ+c : ϕ ∈ X∗, c ∈ R}. For a subset A ⊆ conv(X), sup A stands for the convex function f defined for x ∈ X by f(x) = sup h(x); and (unless stated otherwise) a ∨ b h∈A for the function g defined by g(x)= a(x) ∨ b(x) := max{a(x),b(x)}. By a Minkowski functional p we mean that it is an extended real-valued non-negative sublinear function on X, or equivalently, there is a convex set D ⊆ X with 0 ∈ D so that p is generated by D, i.e. p(x) = inf{λ> 0 : x ∈ λD}, for all x ∈ X. Note that p is l.s.c. if and only if the corresponding D is closed. For example, given a closed subset K ⊆ X, the indicator function δK : x → R∪{+∞} defined by δK(x)=0 for x ∈ K and δK (x)=+∞ for x 6∈ K is a convex proper function on X. In particular, when K =0, then δ0 = δ{0} is a l.s.c. Minkowski functional generated by the singleton {0}. ∗ Given f ∈ conv(X), the subdifferential mapping ∂f : X → 2X of f is defined for x ∈ X by ∂f(x)= {ϕ ∈ X∗ : hϕ, y − xi≤ f(y) − f(x), for all y ∈ X}. By the Brøndsted-Rockafellar theorem [8] (see, also, [12, Theorem 3.18]), for every l.s.c. sublinear function p defined on a Banach space X, we have ∂p(X) := ∂p(x), x[∈X is always nonempty w∗-closed and convex. Obverse that, for a l.s.c. sublinear function p, we have ∂p(X) = ∂p(0). Indeed, for any x∗ ∈ ∂p(X), then, by the definition of ∂p(X), there exists x ∈ X so that hx∗,y − xi≤ p(y) − p(x), ∀y ∈ X. Then, by the subadditivity of p, we have hx∗,zi ≤ p(z + x) − p(x) ≤ p(z) for all z ∈ X. This entails that ∂p(X) ⊆ ∂p(0). Conversely, one can use the support function to convert convex subsets to convex functions. More precisely, for a l.s.c. sublinear function p, we have that ∗ p(x)= σ∂p(0)(x) := sup hx , xi, ∀x ∈ X. x∗∈∂p(0)

For G ⊆ conv(X), we denote Bf (G)= {g ∈ G : g ≤ f}. We simply write Bf = Bf (conv(X)), and Af = Bf (aﬀ(X)).

Since every extended real-valued l.s.c. convex function can be represented as the sup-envelope of a subset of aﬀ(X) (see, for instance [8]), we have sup h = f = sup h. h∈Bf h∈Af

Note that every affine function u = ϕ + c can be regarded as a linear functional ϕ ⊕ c ∈ X∗ ⊕ R restricted to the affine subspace X ⊕{1} of X ⊕ R. We have the following property. Proposition 2.1. For every f ∈ conv(X) there is a l.s.c. sublinear function p defined on X ⊕ R so that f = p(·, 1).

∗ Proof. Note that sup h = f, where Af = {ϕ + c ≤ f : ϕ ∈ X ,c ∈ R}. We deﬁne the sublinear function p for h∈Af (x, r) ∈ X ⊕ R by p(x, r) = sup{hϕ, xi + rc : ϕ + c = ϕ ⊕ c ∈ Af }. Then f = p(·, 1). Deﬁnition 2.1. Let S be a partially ordered set.

i) S is said to be an upper semi-lattice provided for all a,b ∈ S, a ∨ b ∈ S. ii) A subset U of S is called ordered provided for any a,b ∈ U we have either a ≥ b, or b ≥ a. iii) A subset U of S is called an upper bound provided for each s ∈ S there is u ∈ U so that u ≥ s. An upper bound U is called the minimum upper bound if for every upper bounded V of S we have U ⊆ V .

4 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

iv) A subset U ⊆ S is said to be orderless if for every two different elements a,b ∈ U, we have a 6≤ b and b 6≤ a.

Before stating the next lemma, we now introduce the notation [u, v]. For u, v ∈ conv(X), the segment [u, v] is the closed convex set {λu + (1 − λ)v : λ ∈ [0, 1]}. Lemma 2.2. Suppose that X is a Banach space with dim(X) ≥ 2. Let u = ϕ+a, v = ψ+b ∈ aﬀ(X), and h = u∨v. ∗ If ϕ 6= ψ ∈ X , then the segment [u, v] is the minimum upper bound of Ah = {t ∈ aﬀ(X),t ≤ h}.

∗ Proof. We ﬁrst claim that [u, v] is an upper bound of Ah. For φ = x + c ∈ aﬀ(X) so that φ ≤ h, then there exists ξ ∈ R so that h − ξ is a sublinear function on X. Since φ − ξ ≤ h − ξ, then it follows that there exists λ ∈ [0, 1] satisfying x∗ = λϕ + (1 − λ)ψ. Therefore, φ ≤ λu + (1 − λ)v.

Secondly, we claim that [u, v] is the minimum upper bound of Ah. Indeed, for every upper bound V of Ah, and λ ∈ [0, 1], there exists φ ∈ V so that λu + (1 − λ)v ≤ φ. Since φ and λu + (1 − λ)v are afﬁne functions so that φ ≤ h and λu + (1 − λ)v ≤ h = u ∨ v, then φ = λu + (1 − λ)v. It follows that [u, v] ⊆ V . Therefore, [u, v] is the minimum upper bound of Ah.

3 Cones of convex functions and their sup-generating classes

This section is devoted to the study of certain types of convex functions and their sup-generating class. Several concepts will be introduced and properties of these concepts will be discussed. For a real Banach space X, we denote by C(X) the set of all subcones C of conv(X) satisfying f ∨ g ∈ C for all f,g ∈ C, and by C0(X) the set of the following speciﬁc classes of convex functions, which are of our main interest. ∗ Precisely, subl(X), mink(X), semn(X) conv∗(X ) = C0(X) ⊆ C(X), where subl(X) — the cone of all l.s.c. sublinear functions on X; mink(X) — the cone of all l.s.c. Minkowski functionals on X; semn(X) — the cone of all l.s.c. seminorms on X;

∗ ∗ ∗ conv∗(X ) — the cone of all w -l.s.c. convex functions deﬁned on X .

For any ∈ C(X), C(X) presents the subcone of consisting of all continuous functions in . For example, Cconv(X) stands for the cone of all continuous convex functions on X.

Note that C(X) is the set consisting of all subcones of conv(X), hence, for K ∈ C(X), CK is the subcone of all continuous convex functions in K. Denoted by (CK ,̺) equipped with the pointwise convergence topology, i.e. ∗ fα → f in the ̺-topology provided fα(x) → f(x) for all x ∈ X. Note that ̺-topology is equivalent to the w - topology when K = X∗. Definition 3.1. Let C ∈ C(X). A subset G ⊆ C is said to be a sup-generating class of C provided for every f ∈ C there is B ⊆ G so that sup h = f. h∈B To illustrate the definition of sup-generating class, we now provide a concrete example which shows that for certain subcone C ∈ C(X), sup-generating class G could be much more simpler than C itself. Example 3.1. In particular, let C = conv(X) ∈ C(X), then, by an easy application of the Hahn-Banach theorem, we have that the set of all continuous affine functions aff(X) is a sup-generating class for conv(X).

There are many possibilities, but the following collections of sup-generating classes are of main interest to us: ∗ ∗+ ∗ i) G1 = aﬀ(X), ii) G2 = X , iii) G3 = X , iv) G4 = |X |, where X∗+ = {φ ∨ 0 : φ ∈ X∗} and |X∗| = {φ ∨−φ : φ ∈ X∗}. Put

4 G = Gj . j=1 [

5 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

Deﬁnition 3.2. Let C ∈ C(X), and G be a sup-generating class of C.

i) G is called pure provided it is closed under multiplication of non-negative numbers, and Bf (C) ⊆ G for all f ∈ G. ̺ ii) We say that G is atomic if G ⊆ CC and ̺-closed (i.e. for any net {fα} ⊆ G if fα −→ f for some f ∈ CC , then f ∈ G) such that for every S ⊆ G with sup h = g ∈ G there is a monotone non-decreasingnet (gα) ⊆ S h∈S so that lim gα = g in the ̺-topology. α iii) G is said to be perfect if it is purely atomic. Proposition 3.1. Every perfect class G ofa cone C ⊆ conv(X) is the minimum element in all ̺-closed sup-generating classes of C.

Proof. Let F be a ̺-closed sup-generating class of C, and let f ∈ G. Assume that A ⊆ F satisﬁes sup h = f. Then h∈A ̺ the purity and atomic of G entails f ∈ A ⊆ F. Therefore, G ⊆ F . Proposition 3.2. Let X be a Banach space. Then

i) conv(X) admits a perfect class aﬀ(X) ∈ G1(X); ∗ ii) subl(X) admits a perfect class X ∈ G2(X);

∗+ iii) mink(X) admits a perfect class X ∈ G3(X); ∗ iv) semn(X) admits a perfect class |X | ∈ G4(X).

̺ Proof. i). Clearly, aff(X) is a ̺-closed cone. Indeed, assume that {fα} is a net of aff(X) so that fα −→ f for some f ∈ Cconv(X). Then, by the fact that fα are affine for all α, then f is also affine. Since sup h = sup h = f for every f ∈ conv(X), aff(X) is a sup-generating class of conv(X). To see that aff(X) h∈Af h∈Bf is perfect, it suffices to note − Bf = {f + c : c ∈ R }⊆ aff(X), for all f ∈ aff(X). ∗ ∗ ∗ ∗ Hence, if S = {hβ}β ⊆ aff(X) with sup hβ = x + c for some x ∈ X and c ∈ R. By the affinity of x + c, it β ∗ ∗ follows that S ⊂{x + r : r ≤ c}. Without loss of generality, we assume that hβ = x + rβ for all β ∈△ with ∗ ∗ ∗ x + c = sup{x + rβ} = x + sup rβ. β β

∗ ∗ ̺ ∗ Hence, sup rβ = c, which implies that there exists an increasing subnet x + rβ′ so that x + rβ′ −→ x + c. Thus, β perfection of aﬀ(X) follows. ∗ ∗ ii). ̺-closeness of the space X is clear. Indeed, let {xα} be net which ̺-converges to some f ∈ Csubl(X). Then, it follows that f is a continuous linear functional, that is f ∈ X∗. Note that for each f ∈ subl(X), there is a (unique) w∗-closed convex set ∂f(0) = {x∗ ∈ X∗ : x∗ ≤ f} so that sup h = f. This and orderless of X∗ yield purity and atomicity, hence, perfection of X∗. h∈∂f(0)

∗+ ∗+ ∗+ ∗+ iii). Obviously, X is ̺-closed. Indeed, let (xα )α∈△ ⊆ X be a net so that xα is ̺-convergent to some ∗+ ∗ f ∈ Cmink(X). Note that ∂xα (0) = [0, xα] for all α, hence, the fact ∗+ lim inf x = f ∈ Cmink(X), α β≥α β yields that ∗ [0, xβ] ⊆ ∂f(0). α [ β\≥α

6 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

By the Banach-Steinhauss theorem, it follows that ∂f(0) is norm closed. Hence, by passing to a subnet, we can assume ∗ ∗ ∗ that {xα} is norm bounded. Therefore, by the relative w -compactness of bounded subset of X , it follows that there ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗+ exists a further subnet {xα′ } of {xα} so that xα′ w -converges to x0 for some x0 ∈ X . Consequently, f = x0 . To show that X∗+ is purely atomic, it sufﬁces to note that for g, h ∈ mink(X), g ≤ h if and only if ∂g(0) ⊆ ∂h(0). Then, any g ∈ mink(X) so that g ≤ x∗+ , we have that g = λx∗+ for some λ ∈ [0, 1].

∗ ∗ ∗ ∗ ̺ iv). The ̺-closeness of |X | is analogous to the case of iii). Indeed, let {|xα|} ⊆ |X | so that |xα| −→ f for some f ∈ Csemn(X). Then, by the Banach-Steinhauss theorem and the fact that

∗ ∗ [−xβ, xβ] ⊆ ∂f(0), α [ β\≥α ∗ ∗ ∗ ∗ ∗ ∗ it follows that there exists a subnet {xα′ } of {xα} so that xα′ w -converges to x0. Therefore, f = |x0|, which yields ∗ ∗ ∗ that |X | is ̺-closed. Hence, by noting that B|x∗| = {g ∈ semn(X): g ≤ |x |} = {λ|x | : λ ∈ [0, 1]}, the pure atomic of |X∗| follows from a similar argument of iii).

Definition 3.3. Let C ∈ C(X) and A ⊆ C, then A is said to be bounded if sup h(x) < +∞ for all x ∈ X. h∈A Remark. In particular, if C = X∗ and A ⊆ X∗, then by the Banach-Steinhausstheorem, we have that the boundedness defined as above coincides with the norm boundedness of A. Definition 3.4. A cone C ∈ C(X) is said to be (bounded) sup-complete provided for every (bounded) set A ⊆ C we have sup h ∈ C whenever sup h 6= ∞. h∈A h∈A

Given a cone C ∈ C(X), we say that C is a (bounded) sup-completion of C, if C is the smallest cone such that C ⊆ C and C is (bounded) sup-complete. Proposition 3.3. Every cone C ∈ C(X) has a (bounded) sup-completion C.

Proof. It sufﬁces to put C = {sup h 6= ∞ : ∅ 6= A ⊆ C}, h∈A C = {sup h : ∅ 6= A is a bounded set in C} . h∈A We now prove the case that C admits a sup-completion C and the same proof can be applied the case of bounded sup-completion.

Indeed, for any {fα}α ⊆ C with sup fα = g 6= ∞, then, by the deﬁnition of C we have there exist Bfα ⊆ C so that α

sup h = fα for all α. Let A = Bfα , we have that A ⊆ C and g = sup h, which entails that g ∈ C. h∈Bfα α h∈A S Proposition 3.4. i) If the cone C is one of the following four cones conv(X), subl(X), mink(X), semn(X) then C is sup-complete. ii) If the cone C is one of the following four cones

Cconv(X), Csubl(X), Cmink(X), Csemn(X) then C is bounded sup-complete.

Proof. i). If C is one of the four cones as above, then for any family of functions {fα}α ⊆ C, if sup fα is proper, that α is, sup f 6= ∞. Then, sup fα must be a l.s.c. convex function of same type, that is sup fα ∈ C. α α α ii). It is analogous to the proof of ii). Indeed, if C is one of the four cones as ii), then, for any bounded A ⊆ C, we have that sup h(x) < +∞ for all x ∈ X. This entails that sup h is a continuous convex function of same type. h∈A h∈A

7 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

4 Fully order preserving mappings on cones of convex functions

In this section, we shall show that every fully order preserving mapping defined on a bounded sup-complete and ̺- closed cone is bounded ̺-continuous. To begin with, we recall the definition of fully order preserving and reversing mappings. Definition 4.1. Let (P, ≤) be a partially ordered set. A mapping T : P → P is fully order preserving (resp. reversing) if it is a bijection satisfying f ≥ g ⇐⇒ Tf ≥ Tg (resp. f ≥ g ⇐⇒ Tf ≤ Tg). ∗ ∗ ∗ Example 4.1. Let C∗(X ) (resp. C0,∗(X )) be the collection of all nonempty w -closed convex sets (resp. containing ∗ ∗ ∗ the origin 0) of X . Then D = ∂p(0) : subl(X) → C∗(X ) (resp. mink(X) → C0,∗(X )) is a fully order preserving −1 ∗ mapping, where ∂p denotes the subdifferential of p. Conversely, S := D : C∗(X ) → subl(X) defined for ∗ C ⊆ C∗(X ) by −1 (D (C))(x) = suphc, xi = σC (x), x ∈ X c∈C is also fully order preserving. Therefore, for every fully order preserving mapping T : subl(X) → subl(X), −1 ∗ ∗ F := DT D : C∗(X ) → C∗(X ) (4.1) is fully order preserving. The following lemma was motivated by [5, Lemma 2].

Lemma 4.1. Let C ∈ C(X) be a cone, T : C → C be a fully order preserving mapping, and (fα) ⊆ C be a set.

i) If both sup fα and sup Tfα belong to C, then α α

T sup fα = sup Tfα. α α

ii) If, in addition, (fα) is a net, and there is η so that for all ζ ≥ η

lim sup fα, lim sup Tfα, sup fα ∈ C, α α α≥ζ then lim sup Tfα = T lim sup fα. (4.2) α α

Proof. i). Let sup fα = f and sup Tfα = fT . Then f,fT ∈ C. Since T is surjective, then it entails that there is g ∈ C α α so that Tg = fT . Since T is fully order preserving, f ≥ fα implies Tf ≥ Tfα, and further, Tf ≥ Tg. Consequently, f ≥ g. Conversely, Tg ≥ Tfα yields g ≥ fα for all α. Thus, g ≥ f. We have shown f = g. Equivalently, Tf = Tg. ii). Let f = lim sup fα and gα = sup fξ for all α ≥ η. Then the assumption says f,gα ∈ C with gα ≥ f, and α ξ≥α f = limα gα. According to i), Tgα = T sup fξ = sup Tfξ ≥ T f, (4.3) ξ≥α ξ≥α and which further deduces lim Tgα = lim sup Tfα ∈ C. α α Note that T is surjective, then it yields that there is g ∈ C so that lim Tgα = Tg. This and (4.3) entail Tg ≥ Tf. α On the other hand, the isotonicity (i.e. order preserving) of T and non-increasing monotonicity of (gα) deduce that gα ≥ g for all α. Consequently, f = lim sup fα = lim gα ≥ g. This and the isotonicity of T again lead to Tf ≥ Tg. α α We ﬁnish the proof by noting that Tf = Tg is equivalent to (4.2).

Theorem 4.2. Suppose that C ⊆ Cconv(X) is a bounded sup-complete and ̺-closed. If T : C → C is fully preserving. Then T is bounded ̺-continuous, i.e. for every bounded net (fα) ⊆ G̺-convergent to f ∈ G, we have Tf = lim Tfα. α

Proof. Let (fα) ⊆ C be a boundednet which is ̺-convergent to f ∈ C. Since C is a bounded sup-complete, it follows that both g := sup fα and gα := sup fξ belong to C. By Lemma 4.1 i), we have that α ξ≥α

Tf ≤ Tgα = sup Tfξ ≤ Tg ∈ C, for all α. (4.4) ξ≥α

8 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

Note that Tg ∈ C and hence Tg is continuous, then, by the sup-completeness again, it follows that Tgα = sup Tfξ ∈ ξ≥α C for all α.

The non-increasing monotonicity of {Tgα}α implies that the function

gT (x) = lim Tgα(x), x ∈ X, α is a extended real-valued convex function. It follows from (4.4) that

Tf ≤ gT ≤ sup Tfξ ∈ C for all α. (4.5) ξ≥α By the fact that a convex function which is locally bounded above then it must locally Lipschitzian (see [12, p.39 Proposition 3.3]). Thus, by (4.5), we have that gT is continuous. Since C is ̺-closed, then we obtain that gT ∈ C. According to Lemma 4.1 ii), Tf = T lim sup fα = lim sup Tfα. α ξ≥α α ξ≥α

It is easy to observe that for every subnet {fα′ } of {fα} we also have

′ Tf = lim′ sup Tfξ . (4.6) β ξ′≥β′

Therefore, lim Tfα exists and equals to Tf. Indeed, if there exist ε0 > 0, x0 ∈ X and a subnet {fβ′ }β′ of {fα} so α that (Tfβ′ )(x0) − (Tf)(x0) >ε0. By passing to a further subnet, we can assume, without loss of generality, that

(Tfβ′ )(x0) < (Tf)(x0) − ε0. (4.7) Hence, by (4.7), it follows that

′ lim′ sup Tfξ (x0) ≤ Tf(x0) − ε0

conv(X), Cconv(X), subl(X), Csubl(X), mink(X), Cmink(X), semn(X), Csemn(X). Then every fully order preserving mapping is bounded ̺-continuous.

Proof. By Proposition 3.4 ii), we have that

C ∈{Cconv(X), Csubl(X), Cmink(X), Csemn(X)} is bounded sup-complete and the ̺-closeness of C is obvious, then, by Theorem 4.2, it sufﬁces to show the conclusion holds for C ∈{conv(X), subl(X), mink(X), semn(X)}.

Given such a cone C and note that the deﬁnition of ̺-closeness only invokes the subset CC of C. Then, for a fully order preserving mapping T : C → C, according to Theorem 4.2 again, it sufﬁces to prove that the restriction T |CC maps each continuous function f ∈ CC into a continuous one.

Suppose, to the contrary, that there is f ∈ CC but Tf∈ / CC . Since Tf is l.s.c. and proper, there exists 0 6= x0 ∈ X so that

case i). (Tf)(x0)= ∞ if C = conv(X); case ii). (Tf)(λx0)= ∞ for all λ> 0, if C = subl(X), or, mink(X); case iii). (Tf)(λx0)= ∞ for all λ 6=0, if C = semn(X). Let {x0} , case i); D = {λx0 : λ ≥ 0} , case ii); ( {λx0 : λ ∈ R} , case iii).

9 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

For case i) the convex function δD = δx0 is a maximal element of C = conv(X). Since T is fully order preserving,

there is a q ∈ X so that δx0 = Tδq. Since f is continuous, then g := f ∨ δq ∈ C. Therefore, by Lemma 4.1 i),

∞ = Tf ∨ Tδq = Tg ∈ C, and this is a contradiction.

For case ii) the convex function δD is a maximal element but not the maximum in C \{δ0}, TδD is again maximal in C \{δ0}. Therefore, there is 0 6= q ∈ X so that TδE = δD, where E = {λq : λ ≥ 0}. Thus,

δ0 = Tf ∨ TδE = T (f ∨ δE)

is the maximum element in C. But f ∨ δE is not the maximum element of C. This contradicts to that T is fully order preserving. Analogously, we can show that case iii) cannot happen.

5 Fully order preserving mappings restricted to sup-generating classes

We will prove in this section that fully order preserving mappings behave nicely when restricted to certain sup- generating class. Let G be a sup-generating class of C, and a mapping T : G → G. We say that T is affine if for u, v ∈ G with λ ∈ R so that λu + (1 − λ)v, T (u), T (v) and λT (u)+ λT (v) are all in G, then T λu + (1 − λ)v = λT (u)+(1 − λ)T (v). Similarly, we can define a mapping T : G → G is additive (resp. homogeneous) in the same way. The following results state that for a fully order preserving mapping T defined on a cone C, many nice properties of T on a sup-generating class of C can be passed on to the whole cone C. Lemma 5.1. Let C ∈ C(X), and G be a sup-generating class of C. Suppose that T : C → C is a fully order preserving mapping. Then

i) T is afﬁne on C if and only if for all λ ∈ R and u, v ∈ G, T (λu + (1 − λ)v)= λT u + (1 − λ)T v;

ii) T is additive on C if and only if T (u + v)= Tu + T v for all u, v ∈ G;

iii) T is positively homogenous on C if and only if it is positively homogenous on G.

Proof. Given f,g ∈ C and 0 ≤ λ ≤ 1, let (fα), (gβ) ⊆ G such that f = sup fα and g = sup gβ. Then α β

T λf + (1 − λ)g = T λ sup fα + (1 − λ) sup gβ α β = T sup(λfα + (1 − λ)gβ) α, β = sup T λfα + (1 − λ)gβ α, β = sup λT fα + (1 − λ)Tgβ α, β = sup λT (fα) + sup(1 − λ)T (gβ) α β = λT (f)+(1 − λ)T (g), which completes the proof of i). The proof of ii) and iii) are in the same spirt of the case i).

The following proposition states that fully order preserving mappings deﬁned on a class C of convex functions can be reduced into a nice subclass of C. Proposition 5.2. Let C ∈ C(X), and T : C → C be a fully order preserving mapping. Assume that C admits a perfect class G. Then T |G : G → G is again a fully order preserving.

10 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

Proof. Clearly, it sufﬁces to show T G = G. Since G is a sup-generating class and T is a fully order preserving mapping, T G is again a sup-generating class of C. Thus, given g ∈ G, we get sup h = g. Purity of G deduces h∈Bg (TG) Bg(T G) ⊆ G, and further, there is a monotonenon-decreasingnet {gα}α ⊆ T G∩G so that lim gα = lim sup gα = g. α α Let Thα = gα(≤ g) for some hα ∈ G and for each α. By the fact that T is order preserving, then it yields that {hα} is also monotone non-decreasing. Thus, there is a convex function h such that lim hα = h. Now the ̺-closeness of G α implies h ∈ G. By Lemma 4.1 ii),

g = lim Thα = lim sup Thα = T lim sup hα = T h. α α α We have shown G ⊆ T G. Note that T −1 is also fully order preserving. By a similar discussion on T −1, we get T −1(G) ⊇ G. Hence, T G = G.

6 Fully order preserving mappings on certain classes of cones

In this section, we shall show that every fully order preserving mapping defined on some specific classes of cones including conv(X) and subl(X) is affine and ̺-continuous on its sup-generating class. Recall that for two convex functions u, v, denoted by [u, v] the interval generated by u and v, that is [u, v] = {λu + (1 − λ)v : λ ∈ [0, 1]}. A subset M of convex functions is said to be ordered (resp. orderless) if for every u, v ∈ M, then either u ≤ v or v ≤ u (resp. u and v are incomparable). Theorem 6.1. Suppose that C ∈ C(X) is a bounded sup-complete, which consists of continuous convex functions, and T : C → C is a fully order preserving mapping. Assume that C admits a ̺-closed pure class G ⊆ aff(X)= X∗ ⊕ R. Then

i) G is perfect and T : C → C is afﬁne with T G = G; ii) T is ̺-continuous on G; iii) In particular, if G ⊆ X∗ ⊕ R is a w∗-closed linear subspace, then S = T − T (0) is a w∗-continuous linear operator on G.

Proof. i). To prove that G is a perfect class of C, it suffices to show that G is atomic. Assume S ⊆ G so that sup h = g ∈ G ⊆ aff(X). Without loss of generality, g = x∗ + r for some x∗ ∈ X∗, r ∈ R. Since sup h = g, h∈S h∈S ∗ then S must consist of all elements of the form x + rα for some rα ≤ r, and hence sup rα = r. Consequently, let ∗ fβ = x + rβ ⊆ S such that fβ ր g in the ̺-topology. Hence, G is perfect. Applying with the proof of Proposition 5.2, T G = G. In order to show the affinity of T , due to the fundamental theorem of affine geometry and Lemma 5.1, we only need to prove that T [u, v] = [Tu,Tv] for all u, v ∈ G. Note that G ⊆ aff(X), then a segment [u, v] is ordered (resp. orderless) if and only if {u, v} is ordered (resp. orderless). Now, fix two different elements u = ϕ + c, v = ψ + d ∈ G, and let h = u ∨ v. If {u, v} is ordered, say u ≥ v, then h = u = ϕ + c, v = φ + d with c ≥ d. Therefore,

T [u, v] ⊆ BT u(C) ∩{w ∈ C : w ≥ T v} = [Tu,Tv]. Since T −1 is also fully order preserving, by a similar discussion but on T −1, we have T −1[Tu,Tv] ⊆ [u, v]. Thus, T [u, v] = [Tu,Tv]. It remains to the case when {u, v} is orderless, then ϕ 6= ψ, [u, v] and {Tu,Tv} are orderless. By Lemma 2.2, [u, v] is the minimum upper bound of Bh(G). Since T is fully order preserving, T [u, v] is an orderless set which is the minimum upper bound of TBh(G) = BTh(G) = BT u∨T v(G). On the other hand, since [Tu,Tv] is also the minimum upper bound of BT u∨T v(G), we obtain T [u, v] = [Tu,Tv]. Thus, we have shown that T |G is affine. ∗ ii). By Theorem 4.2, we have shown that T |G : G → G is bounded ̺-continuous. Since G ⊆ aff(X) = X ⊕ R, then the ̺-topology can be identified by the w∗-topology by noting Proposition 2.1. By i), we have that T is affine, then S = T − T (0) is linear, which is bounded ̺-continuous. Thanks to the Grothendieck’s dual characterization of completeness (see [14, p.149 Corollary 2]), we know that S : G → G is ̺-continuous, then so does T = S + T (0). iii). It follows from i) and ii) directly.

11 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

Corollary 6.2. Suppose that T : conv(X) → conv(X) is a fully order preserving mapping. Then,

i) the restriction T |Cconv(X) :Cconv(X) → Cconv(X) is bounded ̺-continuousaffine mapping with T aff(X) = aff(X); ∗ ii) S = T |G − T (0) is a ̺-continuous linear operator on X ⊕ R.

Proof. By Theorem 6.1, it sufﬁces to note that for each continuous convex function f, then Tf is also continuous (see, the proof of Corollary 4.3), and note that Cconv(X) is bounded sup-completed and ̺-closed, which admits the perfect class aﬀ(X)= X∗ ⊕ R. Corollary 6.3. Suppose T : subl(X) → subl(X) is a fully order preserving mapping. Then,

∗ ∗ i) the restriction T |Csubl(X) :Csubl(X) → Csubl(X) is bounded ̺-continuous afﬁne mapping with TX = X ;

∗ ∗ ii) S = T |G − T (0) is a w -continuous linear mapping on X .

Proof. The proof is complete analogous to the proof of Corollary 6.2. By Theorem 6.1 it sufﬁces to prove that T maps Csubl(X) onto itself. And this follows from the proof of Corollary 4.3 directly. Furthermore, by Proposition 3.2, we ∗ have that X is a ̺-closed (with respect to Csubl(X)) perfect class of Csubl(X), then, by Theorem 6.1, it follows that T (X∗)= X∗ such that T is w∗-to-w∗ continuous. Theorem 6.4. Let C ∈ C(X) and T : C → C be a fully order preserving mapping. Assume that C admits a ̺-closed ∗+ pure class G ∈ G3, i.e. G ⊆ X . Then G is perfect such that T G = G and T (0) = 0.

Proof. Note for every ϕ ∈ X∗, + Bϕ+ (G)= {λϕ :0 ≤ λ ≤ 1}. (6.1) ∗+ Since G ⊆ X is ̺-closed and pure, it is not difﬁcult to see that G is atomic. By Proposition 5.2, T |G is fully order ∗+ preserving with T G = G. To see that T (0) = 0, it sufﬁces to note that if C admits a perfect class G ⊆ G3 = X , then C is a subclass of Minkowski functionals (i.e. mink(X)) and 0 is the smallest element in mink(X). Then, by the fact that T is a fully order preserving mapping, we have T (0) = 0.

7 Inﬁnite dimensional version of the “Artstein-Avidan-Milman” theorem

In this section, we shall show an exact inﬁnite dimensional version of the “Artstein-Avidan-Milman” theorem for fully order reversing mappings. We begin with the following property. Proposition 7.1. Suppose that X and Y are two Banach spaces. Then the following statements are equivalent:

i) there exists a fully order preserving mapping conv(X) → conv(Y ); ii) X is isomorphic to Y .

Proof. Clearly, it suffices to show i) =⇒ ii). Suppose that T : conv(X) → conv(Y ) is a fully order preserving mapping. Note that aff(X) (resp. aff(Y )) is the perfect sup-generating class of conv(X) (resp. conv(Y )). Since T is fully order preserving, it maps aff(X) onto aff(Y ). Indeed, given φ + r ∈ aff(X), since Bφ+r := {f ∈ conv(Y ): f ≤ φ + r} = {φ + s : r ≥ s ∈ R} is −1 well-ordered, then T (Bφ+r) is again ordered. The only possible case is T (φ + r) ∈ aff(Y ). Since T is also fully order preserving, Taff(X) : aff(X) → aff(Y ) (the restriction of T to aff(X)) is again fully order preserving. By an argument similar to the proof of Theorem 6.1,

Taff(X) : aff(X) → aff(Y ) is a ̺-to-̺-continuous affine isomorphism. Note aff(X)= X∗ ⊕ R, aff(Y )= Y ∗ ⊕ R, and note the ̺-to-̺-continuity is equivalent to the w∗-to-w∗ continuity. Then there is a w∗-to-w∗ continuous isomorphism V : X∗ → Y ∗, which implies that V is an adjoint operator. Hence, there exists an isomorphism U : Y → X so that U ∗ = V .

12 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

The following theorem is an analogous, or, a slight generalization of “Artstein-Avidan-Milman-Iusem-Reem-Svaiter” representation theorem of fully order preserving mappings. However, our proof is simpler. Theorem 7.2. Suppose that X and Y are two Banach spaces. Then for every fully order preserving mapping T : ∗ conv(X) → conv(Y ), there exists an isomorphism U : Y → X, x0 ∈ X, ϕ ∈ Y , α> 0 and r0 ∈ R so that

T (f)(y)= αf(Uy + x0)+ hϕ, yi + r0, f ∈ conv(X), y ∈ Y.

Proof. Suppose that T : conv(X) → conv(Y ) is a fully order preserving mapping. Then by the proof of Proposition 7.1, we obtain that

i) T maps aﬀ(X) onto aﬀ(Y ); ii) S = T − T (0) is ̺-to-̺ continuous fully order preserving linear mapping with S(R)= R, and with S(1) := α> 0.

Since X∗ ⊆ aﬀ(X) is orderless and w∗-closed, S(X∗) is again an orderless w∗-closed hyperplane of aﬀ(Y ) = Y ∗ ⊕ R.

Therefore, there exists y0 + s0 ∈ Y ⊕ R so that ∗ ∗ ∗ ∗ S(X )= {y + s ∈ Y ⊕ R : hy ,y0i + sr0 =0}. ∗ ∗ We claim s0 6=0. Otherwise, we have R ⊆ S(X ). This contradicts to that S(X ) is orderless. Thus, ∗ ∗ ∗ ∗ ∗ S(X )= {y − hy ,y0i/s0 : y ∈ Y }. Let P : Y ∗ ⊕ R → Y ∗ be the projection along {0}⊕ R. Thus, W (x∗)= PS(x∗), x∗ ∈ X∗ ∗ ∗ ∗ ∗ ∗ ∗ is a linear bijection from X onto Y . Indeed, let x1 6= x2, then S(x1) must be incomparable with S(x2). Hence, ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ it follows that W (x1) 6= W (x2). For any y ∈ Y , there exists x + r ∈ X ⊕ R so that S(x + r) = y , which yields that W is surjective. The w∗-to-w∗ continuity of W follows from Lemma 4.1 directly. Therefore, there is isomorphism V : Y → X so that V ∗ = W . Given f ∈ conv(X), let ∗ ∗ Bf aff(X) = {x + r ≤ f : x + r ∈ aff(X)}. Then it follows from Lemma 4.1 for all y ∈ Y (Sf)(y) = S sup h (y) = sup (Sh)(y) (7.1) h∈Bf (aff(X)) h∈Bf (aff(X)) = sup{hSx∗,yi + αr : x∗ + r ≤ f} ∗ ∗ = sup{hW x ,y − y0/s0i + αr : x + r ≤ f} ∗ ∗ = sup{hx , V (y − y0/s0)i + αr : x + r ≤ f} ∗ −1 ∗ = α sup{hx , α V (y − y0/s0)i + r : x + r ≤ f} = αf(Uy + x0),

−1 y0 where U = α V and x0 = −U . Let T (0) = ϕ + r0. Then s0

(Tf)(y)=αf(Uy + x0)+ hϕ, yi + r0, f ∈ conv(X), y ∈ Y.

∗ ∗ Recall that for a Banach space X, conv∗(X ) denotes the cone of all extended real-valued w -l.s.c. convex functions deﬁned on X∗. ∗ Theorem 7.3. Let X be a Banach space. Then there is a fully order preserving mapping from conv(X) to conv∗(X ) if and only if the following two conditions hold:

i) X is reﬂexive; ii) there is a linear isomorphism from X to X∗.

13 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

∗ ∗ ∗ Proof. Sufficiency. Since X is reflexive, we have conv∗(X ) = conv(X ). Let U : X → X be an isomorphism. Then (Tf)(x∗)= f(U −1x∗), f ∈ conv(X), x∗ ∈ X∗, defines a fully order preserving mapping T : conv(X) → conv(X∗). ∗ ∗ Necessity. Suppose that T : conv(X) → conv∗(X ) be a fully order preserving mapping. Note that aff∗(X ) (resp. ∗ ∗ ∗ aff(X)) is perfectclass of conv∗(X ) (resp. conv(X)), where aff∗(X ) is the set of all w -continuous affine functions ∗ ∗ on X . Apply with Lemma 3.2, we have aff(X) is a perfect class of conv(X). It now suffices to show that aff∗(X ) ∗ ∗ ∗ is a perfect class of conv∗(X ). Indeed, aff∗(X ) can be identified by X ⊕ R, which yields the purity of aff∗(X ). ∗ ∗ By the Hahn-Banach theorem, we have aff∗(X ) is a sup-generating class of conv∗(X ). It now suffices to show the ̺ ∗ ∗ ∗ ̺-closeness of aff∗(X ). Indeed, assume that {fα}α∈△ beanetof aff∗(X ) so that fα = xα +rα −→ f ∈ Cconv∗(X ). Then, f is affine, which implies that f − f(0) is a w∗-continuous linear functional. Hence, f − f(0) ∈ X, that is ∗ f = x + r for some x ∈ X and r ∈ R. Therefore, the perfectness of aff∗(X ) has been proven.

By an argument similar to the proof of Theorem 6.1, we obtain that T |aff(X) is a fully order preserving mapping, which is ̺-to-̺ continuous (that is, pointwise-to-pointwise continuous) affine mapping from aff(X) to aff∗(X∗). Let S = T − T (0). Then ∗ ∗ ∗ S|aff(X) : X ⊕ R = aff(X) → aff (X )= X ⊕ R is a fully order preserving ̺-to-̺ continuous linear mapping. This entails that X∗ is isomorphic to X. Note the ̺-to-̺ continuity is equivalent to w∗-to-w continuity in this setting. Then we obtain that X is reflexive. Indeed, T : X∗ → X ∗ is a linear isomorphism which is w -to-w continuous and by the Alaoglu theorem, we have that T (BX∗ ) is a weakly compact subset of X. By the open mapping theorem, it follows that there exists λ> 0 so that λBX ⊆ T (BX∗ ), which implies that BX is weakly compact, hence, yields the reflexivity of X. Corollary 7.4. Suppose that X is a Banach space. Then there is a fully order reversing mapping from conv(X) to itself if and only if X is reflexive and isomorphic to its dual X∗.

Proof. By Theorem 7.3, it suffices to note that if T : conv(X) → conv(X) is fully order reversing if and only if ∗ ∗ FT : conv(X) → conv∗(X ) is fully order preserving, where F : conv(X) → conv∗(X ) denotes the Fenchel transform, which is fully order reversing. Remark. Clearly, for every Hilbert space H, there is a fully order reversing mapping of conv(H). It follows from Corollary 7.4 that there are many non-Hilbert spaces satisfying the existence of fully order reversing mappings of the 1 1 cone of l.s.c. convex functions defined on it. For example, let X = Lp × Lq with 1 < p 6= 2 < ∞, p + q = 1 and endows X with the norm defined for (x, y) ∈ X by k(x, y)k = kxk2 + kyk2. ∗ Then X is reflexive and is isometric to X , which is notp linearly isomorphic to a Hilbert space. Theorem 7.5. Suppose that X is a Banach space. Then for every fully order reversing mapping T from conv(X) onto ∗ ∗ ∗ itself there exist isomorphism U : X → X , x0, ϕ ∈ X , α> 0 and r0 ∈ R so that ∗ (Tf)(x)= α(Ff)(Ux + x0)+ hϕ, xi + r0, for all x ∈ X, ∗ where F : conv(X) → conv∗(X ) is the Fenchel transform.

Proof. Suppose that T : conv(X) → conv(X) is a fully order reversing mapping. Then, FT is a fully order ∗ preserving mapping defined from conv(X) onto conv∗(X ). Applying Theorem 7.2 and Theorem 7.3 we have that ∗ ∗ there exist a w -to-w linear isomorphism V : X → X, x0, v0 ∈ X, α> 0 and β ∈ R such that ∗ ∗ ∗ ∗ ∗ (FTf)(y )= αf(Vy + x0)+ hy , v0i + β, y ∈ X . (7.2) Since the Fenchel transform F satisfies that FF = id, then, by (7.2), we have the following ∗ ∗ ∗ ∗ ∗ (Tf)(x) = (FFTf)(x) = sup hy , xi− αf(Vy + x0) − hy , v0i− β : y ∈ X x − v (7.3) = α sup hy∗, 0 i− f(Vy∗ + x ): y∗ ∈ X∗ − β α 0 Note that V is a linear isomorphism defined from X∗ onto X, then, (7.3) becomes into x − v (Tf)(x)= α sup hV −1ξ, 0 i− f(ξ): ξ ∈ X − hV −1(x ), x − v i− β α 0 0 V −1∗(x − v ) = α sup h 0 , ξi− f(ξ): ξ ∈ X + h−V −1x , xi + (hV −1x , v i− β) α 0 0 0 14 On order preserving and order reversing mappings defined on cones of convex functions MANUSCRIPT

−1∗ −1∗ V ∗ V (v0) −1 −1 Let U = α , x0 = − α , ϕ = −V (x0) and r0 = hV x0, v0i− β, then we get that ∗ (Tf)(x)= α(Ff)(Ux + x0)+ hϕ, xi + r0, for all x ∈ X.

8 An extended fundamental theorem of afﬁne geometry

In the following two sections, we shall discuss representation of fully order preserving mappings defined on the cone semn(X) of all extended real-valued l.s.c. seminorms on a Banach space X. Our main approach is to convert a fully order preserving mapping defined on semn(X) to a lattice isomorphism defined on Csymm,∗(X) (the set of all nonempty w∗-closed symmetric convex subsets of X∗). Then by using the generalized fundamental theorem of affine geometry we obtain a linear isomorphism. To begin with, we recall lattices of convex sets in the following. By a lattice S, we mean that a partially ordered set such that any two of whose elements a and b have a least upper bound a ∨ b and a greatest lower bound a ∧ b. ∗ ∗ ∗ Example 8.1. We order C∗(X ) (the collection of w -closed convex subsets of X ) by inclusion, and define two ∗ operations ∧ and ∨ on C∗(X ) as follows. ∗ w ∗ A ∧ B = A ∩ B, A ∨ B = co (A ∪ B), for all A, B ∈ C∗(X ), (8.1) ∗ w ∗ ∗ where co (A) denotes the w -closed convex hull of A. Then (C∗(X ), ⊆, ∧, ∨) is a lattice. Example 8.2. Let V be a linear space over R. P(V ) stands for the set of all finite dimensional linear subspaces on V . We now define two operations ∧ and ∨ for A, B ∈ P(V ) as follows, A ∧ B = A ∩ B, A ∨ B = span(A ∪ B). Then (P(V ), ⊆, ∧, ∨) is a lattice.

Recall that a mapping ϕ from a lattice S onto itself is a lattice isomorphism if ϕ is bijective and satisﬁes ϕ(A ∧ B)= ϕ(A) ∧ ϕ(B), ϕ(A ∨ B)= ϕ(A) ∨ ϕ(B), ∀A, B ∈ S. The next lemma implies that the fully order preserving mapping F deﬁned by (4.1) is actually a lattice isomorphism ∗ from C∗(X ) onto itself. Lemma 8.1. Let (S, ⊆, ∧, ∨) be a lattice. Then ϕ : S → S is a fully order preserving mapping if and only if it is a lattice isomorphism.

Proof. Sufﬁciency. Suppose that ϕ : S → S is a lattice isomorphism. Let A, B ∈ S with A ≤ B. Note that A ≤ B ⇐⇒ A ⊆ B ⇐⇒ A ∧ B = A. Then ϕ(A)= ϕ(A ∧ B)= ϕ(A) ∧ ϕ(B) ≤ ϕ(B), i.e. ϕ is order preserving. It is fully order preserving since it is bijective and since ϕ−1 is also order preserving. Necessity. Suppose that ϕ : S → S is fully order preserving. Given A, B ∈ S, we know ϕ(A) ⊆ ϕ(A ∨ B) and ϕ(B) ⊆ ϕ(A ∨ B), equivalently, ϕ(A) ∨ ϕ(B) ≤ ϕ(A ∨ B). Since ϕ is surjective, we can choose D ∈ S so that ϕ(A) ∨ ϕ(B) = ϕ(D). Therefore, A ⊆ D and B ⊆ D. Consequently, A ∨ B ⊆ D, and further, ϕ(A ∨ B) ⊆ ϕ(D)= ϕ(A) ∨ ϕ(B). Hence, ϕ(A ∨ B)= ϕ(A) ∨ ϕ(B). We can show ϕ(A ∧ B)= ϕ(A) ∧ ϕ(B) in the same way.

Let V be a linear space over R with dim(V ) ≥ 2. A (V ) stands for the set of all finite dimensional affine subspaces of V and P(V ) for the set of all finite dimensional linear subspaces of V . Now, we define two lattice operations ∧ and ∨ for A, B ∈ P(V ) as follows. A ∧ B = A ∩ B, A ∨ B = span(A ∪ B).

15 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

Then (P(V ), ⊆, ∧, ∨) is a lattice. The following two results (Lemma 8.2 and Theorem 8.3 are called the fundamental theorem of affine geometry when the linear space V is finite dimensional (see, for instance, [1] and [10, pp.56-57]). The fundamental theorem of affine geometry also plays an important role in projective geometry. Lemma 8.2. Suppose that V is a linear space with dim(V ) ≥ 2, and that α : V → V is a mapping. If α fully order preserving from A (V ) onto itself with α(0) = 0, then α is linear bijection.

Proof. Given any u, v ∈ V , let L = span{u, v} = aﬀ{u, v, 0}. Since α is order preserving, it maps every one dimensional subspace of L into a one dimensional subspace of M := α(L). Therefore, α(L)=aﬀ{α(u), α(v), 0} = M. Consequently, α (restricted to L) is a fully order preserving from A (L) onto A (M). Applying the corresponding classical fundamental theorem, we obtain that for all r ∈ R α(u + v)= α(u)+ α(v), α(ru)= rα(u), α(rv)= rα(v). Since u, v are arbitrary, α is a bijective linear map of V onto itself.

Theorem 8.3. Suppose that V is a linear space with dim(V ) ≥ 3. Let π : P(V ) → P(V ) be a fully order preserving mapping. Then there exists a bijective linear mapping λ : V → V so that π(M)= λ(M), for all M ∈ P(V ).

Proof. Let H be a hyperplane of V and a ∈ V \ H. Then V = H ⊕ Ra. Let P(H) be the set of all finite dimensional subspaces of H, and let F = {π(M): M ∈ P(H)}. Then we claim that H′ := {N ∈ F } is a hyperplane of V . Indeed, since π : P(V ) → P(V ) is fully order preserving, it is easy to observe that H′ is a linear subspace of V . Choose any u ∈ V \ H. Then π(Ru) * H′, which entails that H′ is a proper subspaceS of V . Suppose that H′ is not a hyperplane. Then there is a subspace N of V with dim N ≥ 2 so that H′ + N = H′ ⊕ N = V. Choose any two dimensional subspace M of N. Since π−1 is also fully order preserving, π−1(M) is also a two dimensional subspace of V with H ∩ π−1(M)= {0}. This is a contradiction. Now, let a′ ∈ V be so that π(Ra)= Ra′. Denote by A (a + H) (resp. A (a′ + H′)), the set of all finite dimensional affine subspaces of a + H (resp. a′ + H′). We define a mapping α : A (a + H) → A (a′ + H′) by α(S) = span(S) ∩ (a′ + H′), S ∈ A (a + H). Since π is fully order preserving, α must be fully order preserving. Note that for each S ∈ A (a + H), there is L ∈ P(H) so that S = a + L. Then α˜(L) := α(a + L) − a′,L ∈ A (H) defines a fully order preserving mapping α˜ : A (H) → A (H′) satisfying α˜(0) = 0. Applying Lemma 8.2, we obtain that α˜ : H → H′ is linear bijective. Finally, we define λ : V → V by

λ(ra + h) := ra′ +α ˜(h), r ∈ R, h ∈ H. Then the bijective linear mapping λ satisﬁes π(M)= λ(M), for all M ∈ P(V ).

Remark. In finite dimensional spaces, the fundamental theorem of affine geometry has been generalized in various ways in [7] and [16], which contain valuable references and historical remarks of the fundamental theorem. The classical fundamental theorem has been applied to study representation of lattice isomorphisms on set lattices consisting of convex sets in finite dimensional spaces. For detailed information, we refer the reader to [9, 15, 17].

16 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

9 Fully order preserving mappings on the cone of seminorms

This section is devoted to the study of characterizations of fully order preserving mappings defined on cone of l.s.c. seminorms. We denote by mink(X) (resp. subl(X), semn(X)) by the set of all l.s.c. Minkowski (resp. sublinear functions, ∗ ∗ ∗ seminorms) defined on X. C0,∗(X ) stands for the set of all w -closed convex subsets of X containing the origin, ∗ ∗ ∗ ∗ C∗(X ) for the set of all nonempty w -closed convex subsets, and Csymm,∗(X ) forthe set of all w -closed symmetric convex subsets of X∗. ∗ ∗ ∗ If we order C0,∗(X ) and Csymm,∗(X ) by inclusion of sets, and define two operations ∧ and ∨ on C0,∗(X ) and ∗ Csymm,∗(X ) as follows:

∗ A ∧ B = A ∩ B, A ∨ B = cow (A ∪ B), for all A, B, (9.1)

∗ w ∗ ∗ ∗ ∗ where co (A) denote the w -closed convex hull of A, then C0,∗(X ) := (C0,∗(X ), ⊆, ∧, ∨) and Csymm,∗(X ) := ∗ (Csymm,∗(X ), ⊆, ∧, ∨) are lattices. For a convex function f deﬁned on X, let D(f)= ∂f(X), (9.2) the image of f under the subdifferential operator ∂, i.e. D(f)= ∂f(x); and let x∈X ∗ S S(A) = sup hx , ·i := σA. (9.3) x∗∈A

It is clear that for a sublinear function f, then D(f) = ∂f(0) with f = σ∂f(0). And the following propositions are immediately from the deﬁnitions of subdifferential operator and support function. Proposition 9.1. With the notions as above, we have the following properties.

∗ ∗ ∗ i) D : subl(X) → C∗(X ) (resp. mink(X) → C0,∗(X ) or semn(X) → Csymm,∗(X )) is a fully order preserving mapping;

∗ ∗ ∗ ii) S : C∗(X ) → subl(X) (resp. C0,∗(X ) → mink(X) or Csymm,∗(X ) → semn(X)) is a fully order preserving mapping. Remark. By the Proposition 9.1 and the fact that combination of fully order preserving mappings is again fully order preserving, then the following statements are immediately.

i) For any fully order preserving mapping subl(X) → subl(X) (resp. T : mink(X) → mink(X), or, semn(X) → semn(X)), FT := DT S is a fully order preserving mapping, hence, a lattice isomorphism ∗ ∗ ∗ from C∗(X ) (resp. C0,∗(X ) or Csymm,∗(X )) onto itself. ∗ ∗ ∗ ∗ ii) For any fully order preserving mapping T : C0,∗(X ) → C0,∗(X ) (resp. C∗(X ) → C∗(X ) or ∗ ∗ Csymm,∗(X ) → Csymm,∗(X )), FT := ST D is a fully order preserving mapping deﬁned from subl(X) (resp. mink(X) or semn(X)).

For the dual X∗ of a Banach space X, we denote |X∗| := {|x∗| := x∗ ∨−x∗ : x∗ ∈ X∗}, and [X∗] := {[−x∗, x∗]: x∗ ∈ X∗}, where [−x∗, x∗]= {λx∗ : λ ∈ [−1, 1]}. ∗ ∗ Lemma 9.2. Let F : Csymm,∗(X ) → Csymm,∗(X ) be a fully order preserving mapping. Then

∗ ∗ i) its restriction to [X ], F |[X∗] is also fully order preserving from [X ] onto itself; ii) there exists a bijective linear mapping Λ: X∗ → X∗ so that F (M)=Λ(M), for all M ∈ P(X∗);

iii) moreover, we can claim F ([−x∗, x∗]) = Λ([−x∗, x∗]), for all x∗ ∈ X∗.

17 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

Proof. i). Let FS = SF , where S is deﬁned as in (9.3). Then ∗ FS : Csymm,∗(X ) → semn(X) is a fully order preserving mapping. Since for any ﬁxed x∗ ∈ X∗, A := {[−z∗,z∗]: z∗ ∈ [−x∗, x∗]} = {[−λx∗, λx∗]:0 ≤ λ ≤ 1} is an ordered set in [X∗], and

∗ ∗ ∗ ∗ ∗ ∗ ∗ FS (A)= {FS [−z ,z ] : z ∈ [−x , x ]} = {p ∈ semn(X): p ≤ FS ([−x , x ])}

∗ ∗ ∗ ∗ ∗ ∗ is also an ordered set in semn(X), which yields that FS ([−x , x ]) = |y | for some y ∈ X satisfying y =0 if and only if x∗ =0. Hence, F [−x∗, x∗] = [−y∗,y∗]. (9.4) ∗ ∗ ∗ ∗ ∗ By noting that Rx = [−rx , rx ] for all x ∈ X and by (9.4), we have that r>0 S F (Rx∗)= Ry∗ := {ry∗ : r ∈ R}, ∗ ∗ i.e. F maps each one dimensional subspace onto a one dimensional subspace of X . Since F : Csymm,∗(X ) → ∗ ∗ Csymm,∗(X ) is fully order preserving, F |[X∗] is also fully order preserving from [X ] onto itself. ii). It suffices to prove that F must map finite dimensional subspaces to finite dimensional subspaces with the same dimension. It is clear that F must map one dimensional subspaces to one dimensional subspaces. By induction, for any 1 ≤ k ≤ n, we have so that dim(V ) = k if and only if dim(F (V )) = k. Suppose that V is a subspace of X∗ n+1 ∗ n+1 ∗ so that dim(V ) = n +1, then there exists {xi }i=1 such that V = Rxi . Since F is a fully order preserving i=1 n+1 C ∗ WC ∗ ∗ mapping of symm,∗(X ), by Lemma 8.1, F is a lattice isomorphism of symm,∗(X ). Thus, F (V )= F (Rxi ). i=1 It follows that dim(F (V )) ≤ n +1. If dim(F (V )) < n +1 contradicts to the assumption. Note that FW−1 is also a ∗ fully order preserving mapping of Csymm,∗(X ), then dim(V ) = n +1 if and only if dim(F (V )) = n +1. Hence, dim(V )= dim(F (V )) for every finite dimensional subspace V ⊆ X∗. Therefore, by Theorem 8.3,

F (M)=Λ(M), for all M ∈ P(X∗). (9.5) Therefore, we have shown ii). ∗ ∗ iii). Fix any x0 ∈ X \{0}. By (9.5), ∗ ∗ F (Rx0)=Λ(Rx0). ∗ ∗ ∗ ∗ ∗ ∗ ∗ Since F [−x0, x0] ⊆ Λ(Rx0), and since Λ is linear, there exists α> 0 so that F [−x0, x0]= αΛ([−x0, x0]). Without loss of generality, we can assume α =1. Fix any x∗ ∈ X∗ \{0}. ∗ ∗ Case I. If x is linearly independent of x0, then

∗ ∗ ∗ ∗ w ∗ ∗ ∗ ∗ [−x , x ]= Rx co R(x − x0) ∪ [−x0, x0] .

∗ \ Since F is a lattice isomorphism on Csymm,∗(X ) and F is continuous when restricted to ﬁnite dimensional subspaces. Hence,

∗ ∗ ∗ ∗ w ∗ ∗ ∗ ∗ F [−x , x ] = F (Rx ) co R(x − x0) ∪ [−x0, x0]

∗ ∗ \ w ∗ ∗ ∗ ∗ = F (Rx ) co F R(x − x0) ∪ F [−x0, x0]

∗ ∗ \ w ∗ ∗ ∗ ∗ = Λ(Rx ) co Λ R(x − x0) ∪ Λ[−x0, x0]

∗ ∗ \ w ∗ ∗ ∗ ∗ = Λ Rx ∩ co R(x − x0) ∪ [−x0, x0] (9.6) = Λ[−x∗, x∗] ,

18 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

i.e. F [−x∗, x∗] =Λ [−x∗, x∗] . ∗ ∗ ∗ ∗ Case II. If x = βx0 for some β 6=0, by choosing any y0, which is linearly independent of x0, then applying Case I, we obtain ∗ ∗ ∗ ∗ F [−y0,,y0 ] =Λ [−y0,y0 ] . ∗ ∗ ∗ Now, we repeat the previous procedure of the proof of iii), but substitute y0 for x0. Since x is linearly independent ∗ of y0 , we have F [−x∗, , x∗] =Λ [−x∗, x∗] .

Hence, iii) is proven. Consequently, the proof is completed . Lemma 9.3. Suppose T : semn(X) → semn(X) is fully order preserving. Then

i) T maps every continuous seminorm p into a continuous seminorm T (p);

ii) T |Csemn(X) is again fully order preserving from Csemn(X) onto itself, where Csemn(X) denotes the cone of all continuous seminorms on X.

Proof. i). For each p ∈ semn(X) is l.s.c., then it is continuous if and only if p(x) < ∞ for all x ∈ X. Let p ∈ Csemn(X). Suppose, to the contrary, that there is x0 ∈ X so that (Tp)(x0)= ∞. Then (Tp)(rx0)= |r|(Tp)(x0)= ∞ for every 0 6= r ∈ R. Let 0, x ∈ Rx , q(x)= δ (x)= 0 Rx0 ∞, otherwise. Then q ∈ semn(X). Therefore, there is g ∈ semn(X) so that Tg = q. Hence,

q ∨ Tp = Tg ∨ Tp = T (p ∨ g)= δ0.

Since δ0 is the maximum of semn(X), and since T is fully order preserving, we know δ0 is a ﬁxed point of T , i.e. p ∨ g = T (p ∨ g)= δ0. Since f(x) < ∞, ∀x ∈ X, g = δ0, and this contradicts to Tg = q = δRx0 (6= δ0). ii). By i) we have just proven, we see that T |Csemn(X) is order preserving from Csemn(X) into itself. Note that −1 −1 T : semn(X) → semn(X) is also fully order preserving. Then by i) again, we get T |Csemn(X) is order preserving from Csemn(X) into itself. Thus, T |Csemn(X) is again fully order preserving. Theorem 9.4. Suppose that T : semn(X) → semn(X) is fully order preserving. Then there exists a unique isomorphism U : X → X so that Tf(x)= f(Ex), for all f ∈ semn(X), x ∈ X; where E ∈ {±U}.

Proof. Let D (resp. S) be deﬁned as (9.2) (resp. (9.3)). Then by Proposition 9.1, we have that both D and S are fully order preserving mappings. Since the composition of fully order preserving mappings is again fully order preserving, then ∗ ∗ F := DT S : Csymm,∗(X ) → Csymm,∗(X ) (9.7) is fully order preserving. Applying Lemma 9.2 ii), there is a bijective linear mapping Λ: X∗ → X∗ so that F (M)=Λ(M), for all ﬁnite dimensional subspace M ⊆ X∗ and F ([−x∗, x∗]) = Λ([−x∗, x∗]), for all x∗ ∈ X∗. (9.8) Note here that for every w∗-compact symmetric convex subset A, we have that

A = [−x∗, x∗]. ∗ x[∈A We now claim that F (A)= F [−x∗, x∗] (9.9) x∗∈A [ 19 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

∗ for all A ∈ Csymm,∗(X ). Indeed, by the fact that F is order preserving, we have

F [−x∗, x∗] ⊆ F (A). x∗∈A [ Conversely, for any y∗ ∈ F (A), by the fact that F −1 is also order preserving, then it entails that

F −1 [−y∗,y∗] ⊆ A.

Due to Lemma 9.2 i), it follows that there exists z∗ ∈ A with F −1 [−y∗,y∗] = [−z∗,z∗], i.e. F [−z∗,z∗] = [−y∗,y∗]. Note that y∗ is arbitrary in F (A), which yields that F (A) ⊆ F [−x∗, x∗] . x∗∈A [ Combining with (9.8) and (9.9) we have that

∗ F (A)=Λ(A), A ∈ Csymm,∗(X ). Applying Lemma 9.3 ii), T maps each continuous seminorm into a continuous one and note that F is a fully order preserving mapping induced by T . Hence, for every w∗-compact symmetric convex subset A,

Λ(A)= F (A)= ∂(T σA)(0), which entails that Λ maps w∗-compact symmetric convex subsets of X∗ to w∗-compact symmetric convex subsets. By the Banach-Steinhauss theorem, it yields that Λ is continuous in norm. Furthermore, we shall show that Λ is w∗-to-w∗ continuous. Indeed, thanks to the Grothendieck’s dual characterization of completeness (see [14, p.149 Corollary 2]), ∗ ∗ ∗ ∗ ∗ w ∗ w it sufﬁces to show that if {xα}α is a bounded net in X with xα −−→ 0, then Λ(xα) −−→ 0. Note equality (9.8) above is equivalent to T (|x∗|)= |Λ(x∗)|, for all x∗ ∈ X∗. (9.10) Then, apply with Theorem 4.2 and (9.10), it follows that lim |Λ(x∗ )(x)| = lim T |x∗ | (x)=0, α α α α for all x ∈ X, which yields that Λ is w∗-to-w∗ continuous. Consequently, there exists a linear isomorphism U : X → X so that Λ= U ∗. Since f(x) = sup{|hx∗, xi| : x∗ ∈ ∂f(0)} for all f ∈ semn(X) and x ∈ X, (Tf)(x) = sup{T |hx∗, xi| : x∗ ∈ ∂f(0)} = sup{|Λ(x∗)|(x): x∗ ∈ ∂f(0)} = sup{|x∗|(Ux): x∗ ∈ ∂f(0)} = f(Ux). We have proven T (f)(x)= f(Ux), ∀ f ∈ semn(X), x ∈ X. It remains to show that U is a unique. Suppose that there is an isomorphism E : X → X so that T (f)(x)= f(Ex), ∀ f ∈ semn(X), x ∈ X. Then f(Ex)= f(Ux), ∀ f ∈ semn(X), x ∈ X. In particular (by taking f = |x∗|), we obtain |hx∗, Exi| = |hx∗, Uxi|, ∀ x∗ ∈ X∗, x ∈ X. This, in turn, implies E = ±U.

20 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

10 Fully order preserving mappings on the cone of Minkowski functionals

We will show in this section that characterizations of fully order preserving mappings defined on certain classes of convex functions (such as Minkowski functionals, continuous norms, positive homogeneous convex functions of degree p et. al) could be deduced from characterizations of fully order preserving mappings defined on seminorms. Recall that a Minkowski functional is a non-negative positively homogeneous convex function. For a Banach space X, ∗ we denote by mink(X) the cone of all extended real-valued l.s.c. Minkowski functionals defined on X, and C0,∗(X ) is defined as in Section 8, the cone of all w∗-closed convex subsets of X∗ containing the origin. In this section, we shall give a representation theorem for fully order preserving mappings on the cone mink(X). ∗ ∗ ∗ ∗ Denote that [X ]0 := {[0, x ]: x ∈ X }. Theorem 10.1. Let T : mink(X) → mink(X) be fully order preserving. Then

i) T |semn(X) (the restriction of T to semn(X)) is again fully order preserving; ii) there is a unique isomorphism E : X → X so that Tf(x)= f(Ex), ∀f ∈ mink(X), x ∈ X. (10.1)

∗ ∗ Proof. i). Let D : mink(X) → C0,∗(X ) (resp. S : C0,∗(X ) → mink(X) ) be deﬁned as (9.2) (resp. (9.3)), i.e. D(f)= ∂f(0), f ∈ mink(X). ∗ ∗ (resp. S(C)(x)= σC (x) = sup hx , xi, C ∈ C0,∗(X ), x ∈ X.) x∗∈C ∗ ∗ Then F = DT S : C0,∗(X ) → C0,∗(X ) is a fully order preserving mapping with F ({0}) = {0}, the minimal ∗ element of the lattice (C0,∗(X ), ⊆, ∧, ∨). By an argument similar to the proof of Lemma 9.2 i), we get that F maps ∗ [X ]0 onto itself. Next, we show that F ([0, −x∗] ⊆ RF [0, x∗], for all x∗ ∈ X∗. (10.2)

Suppose, to the contrary, that there exists x∗ ∈ X∗ so that if F [0, −x∗] 6⊆ RF [0, x∗]. Then

C := co F ([0, −x∗]) F ([0, x∗]) = F ([0, −x∗]) ∨ F ([0, x∗])

[ ∗ is a convex set but not a segment. Consequently, there is [0,y0] ⊆ C satisfying

∗ ∗ ∗ [0,y0] 6⊆ F ([0, −x ]) F ([0, x ]).

∗ ∗ ∗ [∗ ∗ ∗ ∗ Since F :[X ]0 → [X ]0 is bijective, there exists [0,z0 ] ∈ [X ]0 so that F ([0,z0 ]) = [0,y0]. Since [0,y0 ] ⊆ C, we have that ∗ ∗ ∗ ∗ ∗ [0,z0 ] ⊆ [0, −x ] ∨ [0, x ] = [−x , x ]. It follows that ∗ ∗ ∗ ∗ [0,y0 ]= F ([0,z0 ]) ⊆ F ([0, −x ]) F ([0, x ]), which is a contradiction. Hence, we have shown that F maps lines into[ lines. By induction, it follows that F induces a fully order preserving mapping deﬁned from P(X∗) onto itself. Applying Theorem 8.3 and the proofs in Lemma 9.2 and Theorem 9.4 we have there exists a unique isomorphism U : X → X so that F [−x∗, x∗] = U ∗ [−x∗, x∗] , for all x∗ ∈ X∗. (10.3)

∗ Since Tf = σF (∂f(0)) for all f ∈ mink(X) and [X ] is a sup-generating class of semn(X), then, by (10.3), we get that T |semn(X) is a fully order preserving mapping deﬁned from semn(X) onto itself. Hence, by Theorem 9.4, we have that T |semn(X)f(x)= f(Ex), ∀f ∈ semn(X), x ∈ X, (10.4) where E ∈ {±U}. ∗ ∗ ∗ ∗ ii). By (10.3) and the fact that [X ]0 = {[0, x ]: x ∈ X } is a sup-generating class of mink(X), it follows that there are only the following two possible cases for T : Tf(x)= f(Ux), ∀f ∈ mink(X), x ∈ X, (10.5)

21 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

or Tf(x)= f(−Ux), ∀f ∈ mink(X), x ∈ X. (10.6)

It follows from (10.2) and (10.3) that either F [0, x∗] = U ∗ [0, x∗] , for all x∗ ∈ X∗, or F [0, x∗] = −U ∗ [0, x∗] , for all x∗ ∈ X∗.

Clearly, the former case is equivalent to (10.5), i.e. E = U; and the later case is equivalentto (10.6), i.e. E = −U.

∗ We use norm(X) to denote all (equivalent) norms on X and Cssymm,c,∗(X ) the set of all solid (i.e. with nonempty interior) w∗-compact symmetric convex subsets of X∗. Theorem 10.2. Suppose that X is a Banach space. Then

i) every fully order preserving mapping T : norm(X) → norm(X) is a restriction of a fully order preserving mapping S : semn(X) → semn(X); ii) every fully order preserving mapping T : semn(X) → semn(X) is an extension of a fully order preserving mapping S : norm(X) → norm(X).

Proof. i). Suppose T : norm(X) → norm(X) is a fully order preserving mapping. Let D and S be defined as in ∗ previous sections, i.e. T = DT S, then T is a fully order preserving mapping from Cssymm,c,∗(X ) onto itself. Let us ∗ ∗ ∗ denote by Csymm,c,∗(X ) the set of all symmetric w -compact convex subset of X . We now extend T to be a fully ∗ order preserving mapping from Csymm,c,∗(X ) onto itself. ∗ ∗ w + + More precisely, given any A ∈ Csymm,c,∗(X ), let Aq = A ∨ (qBX∗ ) = co (A ∪ qBX∗ ), q ∈ Q where Q = {r ∈ Q : r> 0}. Then A = Aq. Define q∈Q+ T T˜(A)= T (Aq ). (10.7) + q∈\Q ∗ We claim that T˜ is well defined and it is a fully order preserving mapping defined from Csymm,c,∗(X ) onto itself. ∗ + Indeed, choose any D ∈ Cssymm,c,∗(X ), then there exist λi ∈ Q ,i =1, 2 such that

λ1BX∗ ⊆ D ⊆ λ2BX∗ . ′ Let Aq = A ∨ (qD), then ′ Aλ1q ⊆ Aq ⊆ Aλ2q,

∗ ′ where Aλiq = A ∨ (λiqBX ),i =1, 2. Since T is a fully order preserving mapping, then T Aλ1q ⊆ T Aq ⊆ T Aλ2q, ˜ ′ ˜ which entails that T (A) = F(Aq). This implies that the image of T does not dependent on the choice of q∈Q+ ∗ D ∈ Cssymm,c,∗(X ), hence T˜Tis well deﬁned.

C ∗ + ∗ ′ C ∗ For arbitrary M ∈ symm,c,∗(X ), q ∈ Q , let Mq = M ∨ (qBX ), then there exists Kq ∈ ssymm,c,∗(X ) such that ′ ′ ˜ T (Kq)= Mq. Let K = Kq, we now prove that T (K)= M. q∈Q+ T + ′ C ∗ + ∗ ′ Indeed, for arbitrary q ∈ Q ,Kq ∈ ssymm,c,∗(X ), then there exist λi ∈ Q ,i = 1, 2 such that λ1BX ⊆ Kq ⊆ λ2BX∗ , then ′ Kλ1 ⊆ Kq ⊆ Kλ2 ,

∗ ˜ where Kλi = K ∨ (λiBX ),i =1, 2. Then T (Kλ1 ) ⊆ Mq ⊆ T (Kλ2 ), which implies that T (K)= T (Kλ)= λ∈Q+ −1 Mq = M. Hence T˜ is surjective. Since T is order preserving which implies the injective of T˜.T Therefore, T˜ q∈Q+ ∗ isT a fully order preserving mapping of Csymm,c,∗(X ). Consequently, T = ST D is a fully order preserving mapping from Csemn(X) onto itself. Hence, there exists a unique automorphism U : X → X, so that

(Tf)(x)= f(Ex), for allf ∈ Csemn(X), x ∈ X;

22 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT where E ∈ {±U}. Therefore, there exists a fully order preserving mapping S : semn(X) → semn(X) so that S|norm(X) = T ii). Suppose that T : semn(X) → semn(X) is a fully order preserving mapping. By Theorem 9.4, T is again fully order preserving on norm(X).

Recall that an extended real-valued function f deﬁned on a Banach space X is said to be positively (resp. absolutely) homogeneous of degree p if it satisﬁes f(λx)= λpf(x), ∀λ ≥ 0, x ∈ X. (resp. f(λx)= |λ|pf(x), ∀λ ∈ R, x ∈ X.) We denote by php(X) (resp. ahp(X)), the cone of l.s.c. positively (resp. absolutely) homogeneous convex functions of degree p. Corollary 10.3. Suppose that T : php(X) → php(X) (resp. ahp(X) → ahp(X)) is a fully order preserving mapping. Then there is a unique isomorphism U : X → X so that T (f)(x)= f(Ux), ∀ f ∈ php(X), x ∈ X. (10.8) (resp.T (f)(x)= f(±Ux), ∀ f ∈ ahp(X), x ∈ X.)

Proof. Since a l.s.c. convex function f is positively (resp. absolutely) homogeneous of degree p (1 ≤ p< ∞) if and only if it is of the form f = kp for some l.s.c. Minkowski functional (resp. seminorm) k (see, for instance [13]), ϕ(f 1/p)(x) = (Tf)1/p(x), f ∈ php(X) (resp. ahp(X)), x ∈ X (10.9) deﬁnes a fully order preserving mapping ϕ : mink(X) → mink(X) (resp. semn(X) → semn(X)). By Theorem 10.1 (resp. Theorem 9.4), there is a unique isomorphism U : X → X so that ϕ(f 1/p)(x)= f 1/p(Ux), f ∈ php(X), x ∈ X. (resp. ϕ(f 1/p)(x)= f 1/p(±Ux), f ∈ ahp(X), x ∈ X.) This combining with (10.9) entail that (10.8) holds.

11 Fully order preserving mappings on the cone sublinear functions

We conclude this paper with a representation theorem of fully order preserving mappings deﬁned on cone consisting of sublinear functions, i.e. positively homogenous convex functions. The following theorem is a more general version of the representation theorem of fully order preserving mappings on the cone of sublinear functions. Theorem 11.1. Suppose that C ∈ C(X) is a bounded sup-complete, which consists of continuous convex functions, and T : C → C is a fully order preserving mapping. Assume that C admits a w∗-closed sup-generating class G ⊆ X∗, which is a subspace of X∗. Then there exists a continuous linear mapping U : X → X satisfying U˜ : X/F → X/F (deﬁned by U˜(x + F )= U(x), x + F ∈ X/F ) ∗ is an linear isomorphism, and ϕ0 ∈ X so that

(Tf)(x)= f(Ux)+ hϕ0, xi, for all x ∈ X, (11.1) where F = ⊥G := {x ∈ X : hϕ, xi =0, for all ϕ ∈ G}.

∗ ∗ Proof. Since G ∈ G2 ⊆ G1 is a w -closed subspace of X , G is a perfect generating class of C. By Theorem ∗ 6.2, T is w -continuous afﬁne on C with T G = G. Write T (0) = ϕ0 ∈ G, and let S = T |G − ϕ0. Then S is a fully order preserving w∗-to-w∗ continuous linear operator on G. Put F = ⊥G. Then (X/F )∗ = F ⊥ = G. Let U : X/F → X/F be deﬁned for x by hSϕ, xi = hϕ, Uxi, for all ϕ ∈ G. Then it is easy to see that U is a bounded linear operator with U ∗ = S. Since S is bijective, it yields that U˜ : X/F → X/F is bijective, hence, a isomorphism. Therefore, Sϕ = ϕU for all ϕ ∈ G. Now, given f ∈ C, by Lemma 4.1 i), Sf = sup h = f(U·), h∈BSf (G) that is, (11.1) holds.

23 On order preserving and order reversing mappings deﬁned on cones of convex functions MANUSCRIPT

The following corollary is immediately from Theorem 11.1 by letting C = subl(X). Corollary 11.2. Suppose that T : subl(X) → subl(X) is a fully order preserving mapping. Then there exists a linear ∗ isomorphism U : X → X, and ϕ0 ∈ X so that

(Tf)(x)= f(Ux)+ hϕ0, xi, for all x ∈ X.

Acknowledgements This work was supported by the Natural Science Foundation of China (Grant No. 11731010 & 11371296). The authors would like to thank the referees for their constructive comments and helpful suggestions. The ﬁrst named author is grateful to Professor Shangquan Bu, Professor Chunlan Jiang and Professor Quanhua Xu for their very helpful conversations on this paper.

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