Convexity properties of functions defined on metric Abelian groups

The notions of quasiconvexity, Wright convexity and convexity for functions defined on a metric Abelian group are introduced. Various characterizations of such functions, the structural properties of the functions classes so obtained are established and several well-known results are extended to this new setting.


Introduction
Let X be a linear space and t ∈ [0, 1]. A subset D ⊆ X is termed t-convex if, for all x, y ∈ D, tx + (1 − t)y ∈ D.
As a consequence of a result by Daróczy and Páles [4], every t-convex function (where t ∈ ]0, 1[ ) is automatically Jensen convex and hence Q-convex, i.e., it is r-convex for all rational numbers r ∈ [0, 1] (cf. [11]). The following more general result about t-convexity was established by Kuhn [12].
Theorem A. If D contains at least two points and f : D → R is a t-convex function for some t ∈ ]0, 1[ , then f is s-convex for all s ∈ Q(t) ∩ [0, 1], where Q(t) denotes the smallest subfield of R containing t. Furthermore, for every subfield F of R, there exists a function f : D → R which is t-convex if and only if t ∈ F ∩ [0, 1].
The following result is the multivariable extension of the t-convexity property.
Theorem C. Let D be an open convex subset of a normed linear space and let f : D → R be a Jensen convex function which is bounded from above on a nonvoid open subset of D. Then f is continuous and convex, that is, t-convex for all t ∈ [0, 1].
In the paper [16] the question whether t-Wright convexity implies Jensen convexity was investigated and an affirmative answer was proved if t is a rational number. It was also shown that, for a transcendental t, this implication is not true. Furthermore, it turned out that for some second degree algebraic numbers the answer is positive whereas for some second degree algebraic numbers is negative. Bernstein-Doetsch-type theorems for Wright convex functions were established by Olbryś [29] and by Lewicki [14,15]. On the other hand, in [6] Bernstein-Doetsch-type theorems were proven for quasiconvex functions.
All the above mentioned results motivate to investigate the analogous problems in a more general setting. In our previous paper [5] we have defined the convexity of sets in metric Abelian groups with the help of endomorphisms. The purpose of this paper is to adopt and extend this definition to functions and therefore to investigate the associated notions of quasiconvexity, Wright convexity and convexity. Some of our results will generalize Theorem A and Theorem B and also the above described statements.

Metric Abelian groups and convexity of subsets
In this section we briefly recall the terminology, the notations and all the results from [5] which will be instrumental for our approach.
Let (X, +) be an Abelian group and let E(X) denote the family of all endomorphisms. Then (E(X), +, •) is a ring. Thus, every T ∈ E(X) generates an endomorphism T : E(X) → E(X) defined by T (S) := T • S. For a family T ⊆ E(X) we denote T := { T | T ∈ T}. Finally, I stands for the identity map of X. The multiplication of the elements of X by natural numbers is introduced by 1·x := x, and (n + 1)·x := n·x + x (x ∈ X, n ∈ N).
The mapping π n (x) := n·x is always an endomorphism of X. We say that (X, +) is divisible by n ∈ N if the map π n is a bijection (and hence an automorphism) of X. In this case, for x ∈ X, the element π −1 n (x) is denoted as 1 n ·x. The set of natural numbers n for which X is uniquely divisible by n is a multiplicative subsemigroup of N whose unit element is 1, will be denoted by div(X).
For a subset A ⊆ X and n ∈ N, we say that A is n-convex if {n·x | x ∈ A} = {x 1 + · · · + x n | x 1 , . . . , x n ∈ A}.
For properties of n-convex sets, we refer to the paper [8]. In particular, by [8,Proposition 2], we have that if a set is n-and m-convex, then it is also (nm)-convex.
In the case when (X, +) is equipped with a translation invariant metric d, we say that (X, +, d) is a metric Abelian group. Metric groups are automatically topological groups in which the d-norm · d : X → R is defined as x d := d(x, 0). The subadditivity of · d implies that n·x d ≤ n x d for all x ∈ X and n ∈ N. The equality here, may not be valid.
The smallest number c satisfying this condition is called the d-norm of T and is denoted by T * d . The symbol E d (X) will denote the subring of E(X) of all d-bounded endomorphisms. More generally, for T ⊆ E(X), the symbol T d denotes the d-bounded elements of T. The smallest number c such that n·x d ≤ c x d for all x ∈ X, that is π n * d , will simply be denoted by n * d . For n ∈ N, the measure of injectivity of the map π n is the largest number µ d (n) such that Using these notations, we can now formulate an extension of the celebrated Rådström Cancellation Theorem (cf. [37]) which we proved in [5].
Theorem 2.1. Let (X, +, d) be a metric Abelian group and let n 0 ∈ N such that µ d (n 0 ) > 1. Let A ⊆ X be an arbitrary subset, let B ⊆ X be closed and n 0 -convex subset, and C ⊆ X be a d-bounded nonempty subset such that The d-spectral radius of an endomorphism T ∈ E d (X) is defined as The following result is a generalization of the so-called Neumann invertibility theorem.
Theorem 2.2. Let (X, +, d) be a complete metric Abelian group and let T ∈ E d (X) such that ρ d (T ) < 1.
Then I − T is an invertible element of E d (X), furthermore, Given an endomorphism T ∈ E(X), we say that a subset D ⊆ X is T -convex if, for all x, y ∈ D, This condition is equivalent to the inclusion The class of T-convex subsets of X is denoted by C T (X) in what follows. In the particular case when (X, +) is the additive group of a vector space and T = tI for some t ∈ [0, 1], instead of T -convexity, we briefly speak about t-convexity which is a commonly accepted notion (cf. [12]). If X is a uniquely 2-divisible Abelian group, and T = 1 2 · I, that is, T (x) := 1 2 · x, then T -convex sets will also be termed midpoint convex. One can immediately see that if the group X is divisible by some n ∈ N and T = 1 n ·I, then T -convexity is equivalent to n-convexity defined in the previous section. It is obvious but useful to observe that a subset D ⊆ X is T -convex if and only if, for all p ∈ D, Now, given a nonempty subset D ⊆ X, we consider the collection of endomorphisms T of X that make D to be T -convex: It is obvious that, for every set D, we have 0, I ∈ T D and 0, I ∈ T d D (if X is a metric Abelian group). The next result describes a convexity property of T D .
In particular, these sets are closed with respect to the composition of maps.
In the next result, we provide conditions ensuring that T -convexity implies midpoint convexity.
Theorem 2.5. Let (X, +, d) be a complete metric uniquely 2-divisible Abelian group and T ∈ E d (X) such that ρ d (2·T − I) < 1. Then, for every nonempty T -convex set D ⊆ X, the set cl(T d D ) is a midpoint convex subset of E d (X). Furthermore, every closed T -convex subset of X is also midpoint convex.
The following result will be instrumental when investigating T -Wright convex functions. Corollary 2.6. Let (X, +, d) be a metric Abelian group, let n 0 ∈ N such that µ d (n 0 ) > 1 and let D be a closed bounded n 0 -convex set. Let n ∈ N and T 1 , . . . , T n ∈ T d D be such that T : − y, which means that This proves the T -quasiconvexity of f when level sets are T -convex. Finally, assume that we are given a set S ⊆ X and T ∈ T. Note that T -quasiconvexity of −χ S can be directly rewritten as follows: for all x, y ∈ X. This inequality in turn is equivalent to (use the definition of characteristic function): which is precisely the T -convexity of the set S.
Proof. The proof of (i) is obvious. Proof of (ii). To prove the T-quasiconvexity of the pointwise supremum of a family {f α | α ∈ I} of T-quasiconvex functions defined on D fix T ∈ T, x, y ∈ D and ε > 0.
To show that f is T -quasiconvex observe that there exists some α 0 ∈ I such that Since ε > 0 is arbitrary small, then f is T -quasiconvex.
To justify the T-quasiconvexity of the pointwise infimum of a chain {f α | α ∈ I} of T-quasiconvex functions defined on D fix T ∈ T, x, y ∈ D and ε > 0.
To show that f is T -quasiconvex observe that there exist some α x , α y ∈ I such that f αx (x) ≤ f (x) + ε and f αy (y) ≤ f (y) + ε. Since family {f α | α ∈ I} forms a chain, there exists α 0 ∈ {α x , α y } such that f α 0 = min(f αx , f αy ) and then we have Again, since ε > 0 is arbitrary small, we obtain that f is T -quasiconvex.
The resulted equality proves that f is T -quasiconvex.
Proof of (iii). For the T-quasiconvexity of the function f ⋄ g, let x, y ∈ D + E. We need to prove, for all T ∈ T, that Let T ∈ T be fixed. Then, using the definition of f ⋄ g and the T-quasiconvexity of f and g, we obtain Upon taking the limits c ց (f ⋄ g)(x) and d ց (f ⋄ g)(y), the inequality (6) follows. Proof of (iv). To verify the T-quasiconvexity of the function f • A, let x, y ∈ A −1 (D) and let T ∈ T be fixed. Then A(x), A(y) ∈ D, hence the T -quasiconvexity of f yields Finally, we show the T-quasiconvexity of the function f • A −1 . For this proof, let x, y ∈ A(D). For the proof of the inequality , and the T -quasiconvexity of f , we get , the inequality (7) follows.
Using assertion (ii) of Theorem 3.2, it follows that, for every function f : is the largest T-quasiconvex function which is not greater than f on D. This function will be called the T-quasiconvex envelope of f . Now, given a function f : D → [−∞, +∞[ , we consider the collection of endomorphisms T ∈ E(X) that make f to be T -quasiconvex: It is obvious that, for every function f , we have 0, I ∈ T f and 0, I ∈ T d f (if (X, +, d) is a metric Abelian group). The next result shows some structural properties of T f and T d f . Consequently, This means that f is S-quasiconvex, hence S ∈ T f . This yields that T f is T -convex for all T ∈ T f , which was to be proved. The proof of the second assertion is completely analogous.
The next result follows from Theorem 3.3 exactly in the same manner as Corollary 2.4 was deduced from Theorem 2.3. In the following statement we show that T -quasiconvexity implies the midpoint quasiconvexity under certain conditions on T , f , and X.
Proof. In view of Theorem 2.5, we have that D is a midpoint convex set. Let f ∈ Q T (D) and define the sequence of endomorphisms T n by By induction, one can see that this sequence satisfies the recursion . Then, by the last assertion of Corollary 3.4, it follows that T n ∈ T d f for all n ∈ N. The condition ρ d (2·T − I) < 1 implies that T n converges to 1 2 ·I. If R, S ∈ cl(T d f ), then there exist sequences R n and S n in T d f converging to R and S, respectively. By Theorem 3.3, for all n ∈ N, we have that To complete the proof, assume that f is also a lower semicontinuous function. To prove its midpoint quasiconvexity, let x, y ∈ D. Then the midpoint convexity of the set cl(T d f ) and 0, I ∈ cl(T d f ) imply that Upon taking the limit n → ∞ and using the lower semicontinuity of f , it follows that The following result presents a further invariance property of T d f . Theorem 3.6. Assume that (X, +, d) is a metric Abelian group, n 0 ∈ N is such that µ d (n 0 ) > 1 and D is a closed set. Let f : D → [−∞, ∞[ be a lower semicontinuous function whose level sets D c f are bounded n 0 -convex for all c ∈ R. Let n ∈ N and T 1 , . . . , T n ∈ T d f be such that T := T 1 + · · · + T n is a bijection with T −1 ∈ E d (X). Then, for all k ∈ {1, . . . , n − 1}, we have The lower semicontinuity of f implies that the level sets D c f are closed bounded n 0 -convex subsets of the closed set D for all c ∈ R. In view of Proposition 3.1, it follows that these level sets are T 1 -, . . . , T n -convex. Now, applying Corollary 2.6, we obtain that all these level sets are T −1 • (T 1 + · · · + T k )convex. Hence, again by Proposition 3.1, we get that f is T −1 • (T 1 + · · · + T k ) -quasiconvex.

T -Wright convex and T -Wright affine functions
Assume that (X, +) is an Abelian group. For an endomorphism T ∈ E(X), we say that a function is the additive group of a linear space and f is t·I-Wright convex for some t ∈ [0, 1], then we say that f is t-Wright convex. If f is t-Wright convex for all t ∈ [0, 1], then it is called a Wright convex function (in the standard sense) (cf. [39]). If (X, +) is a uniquely 2-divisible group, then 1 2 ·I-Wright convex functions are called Jensen convex (cf. [11]). (ii) If D is a T-convex set, then W T (D) is closed with respect to the pointwise chain supremum, the pointwise chain infimum, and the pointwise convergence. (iii) If an endomorphism A ∈ E(X) commutes with any member of T and f ∈ W T (D), Proof. The proof of (i) is obvious. Proof of (ii). To prove the T-Wright convexity of the pointwise supremum of a nondecreasing family Since the family {f α | α ∈ I} forms a chain, there exists Since ε > 0 is arbitrary small, the T -Wright convexity of f follows. Similarly we will establish the T-Wright convexity of the pointwise infimum of a chain and then we have To show the T-Wright convexity of the pointwise limit (f n ) of T-Wright convex functions defined on D, fix T ∈ T and x, y ∈ D. We have The resulted inequality proves that f is T -Wright convex.
Proof of (iii). To verify the T-Wright convexity of the function f • A, let x, y ∈ A −1 (D) and let T ∈ T be fixed. Then A(x), A(y) ∈ D, hence the T -Wright convexity of f yields It is obvious that, for every function f , we have 0, I ∈ TW f and 0, is a metric Abelian group). The next result shows some structural properties of TW f and TW d f .  (4). If (X, +, d) is a metric Abelian group, then TW d f is also closed with respect to the mappings in (4).
Proof. The invariance of TW f with respect to the map T → I − T is an obvious consequence of the definition.
Let T, S ∈ TW f . Then, for all x, y ∈ D, we have This means that f is (T • S + (I − T ) • (I − S))-Wright convex, which was to be proved. The proof of the second assertion is completely analogous.
The next statement is a generalization of the third assertion of Theorem 1 of the paper [16], which was one of the main results therein. Our approach extensively uses Corollary 2.6 which is based on Theorem 2.1, our generalization of the Rådström Cancellation Theorem. and Therefore, the T -Wright convexity of f implies that Adding up these inequalities side by side with respect to i ∈ {0, . . . , n − 1}, we get f (u 0,j+1 ) + f (u n,j ) ≤ f (u 0,j ) + f (u n,j+1 ).
Now adding up the inequalities side by side with respect to j ∈ {0, . . . , k − 1}, we arrive at the inequality Observe that u 0,0 = x, u n,k = y, and Therefore, inequality (8) shows that f is S −1 • (n·T )-Wright convex, which was to be proved.
In the following statement, we provide conditions for the invertibility of the map S = n·T + k·(I − T ).
Theorem 4.4. Let (X, +, d) be a complete metric Abelian group with µ d (2) > 1, let n, k ∈ N such that n + k ∈ div(X) and µ d (n + k) > 0, let D be a closed bounded 2-convex set, and let f : For the proof of this statement, in view of Theorem 4.3, it suffices to show that inequality (9) implies the invertibility of S with a d-bounded inverse. We will prove this by using Theorem 2.2.
First observe that Therefore, by the subadditivity of the d-norm and the submultiplicativity of µ d , we obtain Now, taking the m-th root side by side, then computing the upper limit as m → ∞, finally using (9), we arrive at the inequality Therefore, Theorem 2.2 applied for the endomorphism I − 2 n+k ·S yields that I −(I − 2 n+k ·S) = 2 n+k ·S is an invertible endomorphism with a bounded inverse. Thus π 2 must be a surjection and hence 2 ∈ div(X). Consequently, 1 n+k ·S = 1 n+k ·I •S is also an invertible endomorphism with a bounded inverse. Therefore, The following result is an easy consequence of Theorem 4.4 and it is still more general than the third assertion of Theorem 1 of the paper [16].
If (X, +) is the additive group of a Banach space, then div(X) = N, µ d (n) = n for all n ∈ N and convex sets are 2-convex.
If t = 1, then (10) and Theorem 2.2 imply that I − (I − 2·T ) = 2·T is invertible with a bounded inverse, hence S = T is invertible with S ∈ E d (X). The inclusion S −1 • (t·T ) = I ∈ TW d f is trivial. The case t = 0 can analogously be seen.
In the rest of the proof, we may assume that t ∈ ]0, 1[ ∩Q. Then there exist n, k ∈ N such that t = n n+k . Then inequality (10) becomes (9), hence, by Theorem 4.4, we get that (n + k)·S = n·T + k·(I − T ) is invertible with a bounded inverse and  [26]).
Theorem 4.6. Let T ⊆ E(X) and assume that there exists T 0 ∈ T such that T 0 (X) = (I −T 0 )(X). Then a function a : X → R is T-affine if and only if there exist a constant c, an additive function A : X → R and a symmetric biadditive function B : X × X → R such that and Proof. Assume first that a : X → R is T-Wright affine. Then, using that a is real valued, by (11) we have that In Therefore, the function a satisfies the functional equation (11) if and only if it has the representation (13) and B(T (x − y), (I − T )(x − y)) = 0 holds for all x, y ∈ X, that is, if condition (12) is valid.
In the paper [3] the functional equation (11) was considered under the assumption that T is given as a multiplication by t ∈ [0, 1] and the characterization of those numbers t was obtained for which there exists a nontrivial biadditive function B satisfying (12).

(T, t)-convex and (T, t)-affine functions
Assume that (X, +) is an Abelian group. For an endomorphism T ∈ E(X) and t ∈ [0, 1], we say that a function f : D → [−∞, +∞[ is (T, t)-convex if D is a T -convex subset of X and, for all x, y ∈ D, Here and in the rest of the paper, we use the usual convention 0 · (−∞) = 0.
The class of R-convex, in particular, (T, τ )-convex functions defined on D are denoted by C R (D) and C T,τ (D), respectively. If (X, +) is the additive group of a linear space and f is (t·I, t)-convex for some t ∈ [0, 1], then we say that f is t-convex. If f is t-convex for all t ∈ [0, 1], then it is called a convex function (in the standard sense). If (X, +) is a uniquely 2-divisible group, then ( 1 2 ·I, 1 2 )-convex functions are exactly the Jensen convex ones. One can observe that the function f ≡ −∞ is trivially (T, t)-convex for arbitrary (T, t) ∈ E(X)×[0, 1]. On the other hand, it is possible that a (T, t)-convex function can take both finite and infinite values. To exclude this possibility, the following lemma will be useful. In what follows, for a T -convex set D, we say that an element p ∈ D is T -internal with respect to D, if D has no proper subset E which contains p and, for all x, y ∈ D with T (x) + (I − T )(y) ∈ E implies x, y ∈ E.
Proof. Assume that f is a (T, t)-convex function which is finite at some T -interior point p. Define the sequence of sets (D n ) by the recursion Observe that p ∈ D 1 , which implies D 0 ⊆ D 1 , and hence (D n ) is an increasing sequence of sets. Let E := ∞ n=0 D n . Then p ∈ E, and taking the union of both sides in (15), it follows that Therefore, E is a subset of D which contains p and, for all x, y ∈ D with T (x) + (I − T )(y) ∈ E implies x, y ∈ E. By the T -internality of p, it follows that E = D, that is, D = ∞ n=0 D n . In the rest of the proof, we show that f is finite-valued on D n for all n ≥ 0. This is obvious for n = 0 by the choice of p. Now assume that f is finite-valued on D n−1 for some n ∈ N. Let z ∈ D n . Then there exist x, y ∈ D such that z ∈ {x, y} and T (x) + (I − T )(y) ∈ D n−1 . Then f (T (x) + (I − T )(y)) > −∞ and, by the (T, t)-convexity of f , (14) holds. The left hand side being finite, the condition t(1 − t) > 0 implies that f (x) as well as f (y) are also finite, which yields that f is finite at z. This completes the induction and finally shows that f is finite valued on D.
Lemma 5.2. Let T ∈ E(X) such that either T (X) ⊆ (I − T )(X) or (I − T )(X) ⊆ T (X) holds. Then every element of X is T -internal with respect to X.
Proof. Let p ∈ X and let E be a set which contains p and, for all x, y ∈ D with T (x) + (I − T )(y) ∈ E implies x, y ∈ E. Then We show that D 1 = X, which shows that E cannot be proper and hence p must be T -internal with respect to X.
Assume that T (X) ⊆ (I − T )(X) holds and let x ∈ X be arbitrary. Then T (x − p) ∈ (I − T )(X), therefore, there exists y ∈ X such that T (x − p) = (I − T )(y). Hence, T (x) + (I − T )(y) = T (p) + (I − T )(p) = p, which shows that x ∈ D 1 . Thus, in this case we have obtained that D 1 = X. In the other case, the argument is completely analogous.

The epigraph of an arbitrary function
For T ∈ E(X) and t ∈ R, the endomorphism (T, t) ∈ E(X × R) is defined as Therefore, any relation R ⊆ E(X) × R can be viewed as a subset of E(X × R) as well. For R ⊆ E(X) × R, we introduce the domain and codomain of R as follows The following characterization of R-convexity of functions is important. Proof. To prove that the epigraph of an R-convex function f is an R-convex set, let (T, t) ∈ R. Then D is a T -convex subset of X. Fix some p, q ∈ epi(f ). There exist x, y ∈ D and u, v ∈ R such that p = (x, u), q = (y, v) and f (x) ≤ u, f (y) ≤ v. From the T -convexity of D and from the (T, t)-convexity of f , we get

This inequality gives us
To prove the converse implication, let (T, t) ∈ R and x, y ∈ D. We have (x, f (x)), (y, f (y)) ∈ epi(f ). Therefore, by the assumed (T, t)-convexity of epi(f ), which yields the T -convexity of D and the (T, t)-convexity of f , and completes the proof.
Proof. The proof of (i) is obvious. Proof of (ii). To verify the R-convexity of the pointwise supremum of a family {f α | α ∈ I} of R-convex functions defined on D, fix (T, t) ∈ R, x, y ∈ D and ε > 0.
To show the R-convexity of the pointwise limit (f n ) of R-convex functions defined on D, let (T, t) ∈ R and x, y ∈ D. We have This inequality proves that f is (T, t)-convex.
Proof of (iii). For the R-convexity of the function f * g, let (T, t) ∈ R and x, y ∈ D + E. We need to prove that Let c ∈ ](f * g)(x), +∞[ and d ∈ ](f * g)(y), +∞[ be arbitrary. Then, by the definition of f * g, there exist u, v ∈ D such that Then, by the (T, t)-convexity of f and g, we obtain Upon taking the limits c ց (f * g)(x) and d ց (f * g)(y), the inequality (16) follows. Proof of (iv). To verify the R-convexity of the function f • A, let x, y ∈ A −1 (D) and (T, t) ∈ R. Then A(x), A(y) ∈ D, hence the (T, t)-convexity of f yields which proves the (T, t)-convexity of f • A.
Finally, we show the R-convexity of the function f • A −1 . For this proof, let x, y ∈ A(D) and (T, t) ∈ R. For the proof of the inequality Using the equality A(T (u)+(I −T )(v)) = T (x)+(I −T )(y) and the (T, t)-convexity of f , we get Upon taking the limits c ց (f • A −1 )(x) and d ց (f • A −1 )(y), the inequality (17) It is easy to see that, for all T ∈ E(X), the set R f (T ) is a closed (possibly empty) subinterval of [0, 1]. Obviously, 0 ∈ R f (0) and 1 ∈ R f (I) for every function f : D → [−∞, +∞[ . If f is nonconstant, then the equalities R f (0) = {0} and R f (I) = {1} can easily be seen.
It is obvious that, for every function f , we have 0, I ∈ dom(R f ) and 0, I ∈ (dom(R f )) d (if (X, +, d) is a metric Abelian group). The next result shows some structural properties of dom(R f ) and (dom(R f )) d .
In particular, these sets are closed with respect to (componentwise) multiplication.
The following characterization of R-affinity is important. Proof. To prove that the graph of an R-affine function a is an R-convex set, let (T, t) ∈ R. Then D is a T -convex subset of X. Fix some p, q ∈ graph(a). Then there exist x, y ∈ D such thatp = (x, a(x)), q = (y, a(y)). From the T -convexity of D and from the (T, t)-affinity of a, we have (18), which gives us = T (x) + (I − T )(y), a(T (x) + (I − T )(y)) ∈ graph(a), which shows that graph(a) is (T, t)-convex.
To prove the converse implication, let (T, t) ∈ R and x, y ∈ D. We have (x, f (x)), (y, f (y)) ∈ graph(a). Therefore, by the assumed (T, t)-convexity of graph(a), T (x) + (I − T )(y), ta(x) + (1 − t)a(y) = (T, t)(x, a(x)) + (I − T, 1 − t)(y, a(y)) ∈ graph(a), which yields the T -convexity of D and the (T, t)-affinity of a, and completes the proof. Proof. The proof of (i) is obvious and the proofs of (ii) and (iii) are parallel to the corresponding statements of Theorem 5.4, therefore, they are omitted. Now, given a : D → [−∞, +∞[ , we consider the set of those pairs (T, t) ∈ E(X) × [0, 1] such that a is a (T, t)-affine function: Obviously, S a ⊆ R a and, for all T ∈ E(X), the set S a (T ) is a closed (possibly empty) subinterval of In particular, these sets are closed with respect to (componentwise) multiplication.
Proof. Let (T, t), (T 1 , t 1 ), (T 2 , t 2 ) ∈ S a . Set Then, a is (T 1 , t 1 )-, (T 2 , t 2 )-and (T, t)-affine, therefore, for all x, y ∈ D, we have Consequently, This means that a is (S, s)-affine, which was to be proved. The proof of the second assertion is analogous. The last assertion easily follows from the first two by taking (T 2 , t 2 ) = (0, 0) in the above proof. This proves that a is (S * • T, s −1 t)-affine.
To show the last assertion, observe that a is (I − S * • T, 1 − s −1 t)-affine and Thus, the statement follows from Theorem 5.8.
It follows from the above theorem that if S * : (I − T )(X) → X is a homomorphism which is the right inverse of S on the codomain of I − T and s + t ≥ 1, then a is also (S + T − I, s + t − 1)-affine. Indeed, Using that (I − T, 1 − t) ∈ S a , the above equality and the last assertion of Theorem 5.9 imply that a is also (S + T − I, s + t − 1)-affine.
The following theorem offers a characterization of R-affine functions defined on X. The main tool for the proof is the result of Székelyhidi [38], which describes the general solution of linear functional equations with constant coefficients.
Proof. Assume that a is an R-affine function which is not identically equal to −∞. First we show that a(x) ∈ R for all x ∈ X. There exists p ∈ X such that −∞ < a(p), i.e., a(p) ∈ R. By Lemma 5.2, it follows that p is T 0 -internal with respect to X, therefore, by Lemma 5.1, a is finite everywhere.
In the case when T 0 (X) ⊆ (I − T 0 )(X) holds, we can rewrite the (T 0 , t 0 )-affinity of a in the following form This is a particular case of the linear functional equations investigated by Székelyhidi. Therefore, by [38,Theorem 3.6], it follows that a is a first-degree generalized polynomial, that is, a = A + c, where A ∈ E(X) and c ∈ R. Now the R-affine property of a implies that holds for all (T, t) ∈ R. Putting y = 0 in this equality, it follows that A is (T, t)-homogeneous. The case when (I − T 0 )(X) ⊆ T 0 (X) is valid, is completely analogous.
The following characterization of R-convex functions is based on Rodé's celebrated separation theorem [36]. A relation R ⊆ E(X) × [0, 1] will be called nonsingular if (T, 0) ∈ R implies T = 0 and (T, 1) ∈ R implies T = I. and Proof. Assume first that, for all p ∈ D, there exists an R-affine function a : D → [−∞, +∞[ satisfying (19). Then, f is the pointwise supremum of R-affine and hence R-convex functions. Thus, by the second assertions of Theorem 5.4, it is an R-convex function.
To prove the other implication, assume that f is R-convex and, for an endomorphism T ∈ dom(R), define the binary operation ω T : X 2 → X by ω T (x, y) := T (x) + (I − T )(y).
Then, ω T is idempotent and, by the dom(R)-convexity of D, it follows that D is closed with respect to the operation ω T for all T ∈ dom(R). The assumption that dom(R) forms a pairwise commuting family of endomorphisms, easily implies that the set of operations {ω T | T ∈ dom(R)} is also commuting in the following sense: Therefore, all basic assumptions of the theorem of Rodé are satisfied. The R-convexity of f is now equivalent to the property Let now p ∈ D be fixed and define g : Then, by the idempotent property of the operation ω T and by the nonsingularity of the relation R, we can see that g satisfies the inequality, i.e., g is R-concave. In addition, we trivially have that g ≤ f on D. Thus, by the theorem of Rodé, it follows that there exists a function a : D → [−∞, +∞[ between g and f which satisfies the equality The following result is a direct consequence of Theorem 5.11 and Theorem 5.12.
Applying the inverse endomorphism S side by side to this equality and again using the commuting property of the endomorphisms, it follows that , . . . , n}).
From here, we can see that c n = (1 − t n−1 )c n−1 , which proves equality (25) in the case i = n. Finally, assume that i ∈ {k + 1, . . . , n − 1}. Then After these preparations, multiply the inequality (23) by c i side by side and sum up the resulting inequalities for i ∈ {1, . . . , n}. Then, in view of the equalities (25), the terms containing f (u i ) for i ∈ {1, . . . , k − 1, k + 1, . . . , n} cancel out and we obtain This is equivalent to the inequality Observe that in each inequality of (23), the sums of the coefficients on the left and right hand side are equal to each other. This remains valid after multiplying by c i and summing up the inequalities so obtained. In particular, this has to be true for the inequality (26). As a result, it follows that c k − (1 − t k−1 )c k−1 − t k+1 c k+1 = 1. Hence, the inequality (26) proves that f is (R k , r k )-convex.
Proof. For i ∈ {1, . . . , n}, define Then, these endomorphisms and constants satisfy all the assumptions of the previous theorem, furthermore, the equalities in (20) are satisfied. Therefore, the conclusion of this result applies.
The next corollary is a generalization of former results of Daróczy-Páles [4] and Kuhn [12] which are related to the vector space setting.