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Convexity of sets in metric Abelian groups

In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups equipped with a translation invariant metric, we define the boundedness, the norm, the measure of injectivity and the spectral radius of endomorphisms. Beyond the investigation of their properties, our first main goal is an extension of the celebrated R\aa dstr\"om Cancellation Theorem. Another result generalizes the Neumann Invertibility Theorem. Next we define the convexity of sets with respect to a family of endomorphisms and we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex and we establish the basic properties of this family. Our first main result establishes conditions which imply midpoint convexity. The next main result, using our extension of the R\aa dstr\"om Cancellation Theorem, presents further structural properties of the family of endomorphisms that make a subset convex.


Introduction
The notion of convexity is usually defined in the context of linear spaces. If X is a linear space, then a subset D ⊆ X is termed convex if, for all x, y ∈ D and t ∈ [0, 1], In other words, D is convex if it is closed with respect to two-term convex combinations. This notion has been generalized in several ways, for instance, given a fixed number t ∈ [0, 1], a set D is called t-convex if the inclusion (1) is valid for all x, y ∈ D. In particular, the 1 2 -convex sets are said to be midpoint convex. It is clear that these classes of subsets are closed with respect to intersection and therefore, one can naturally define also the convex, t-convex and midpoint convex hulls of sets, this is the subject of several papers, e.g. [12,17]. The structural properties of the set of numbers t ∈ [0, 1] such that D is t-convex were intensively discussed, see [6] and some references therein. The notion of convexity in metric spaces was introduced by Menger [9], see also [8], approximate convexity properties of sets were investigated in [1,2,3]. The general theory of convex structures was developed in the book [16]. A possible approach to introduce convexity in topological, particularly in Lie groups is given by Rådström [14].
Two notions of convexity (which are generalizations of Q-convexity) have been introduced in the paper [5] in the Abelian semigroup setting. There, the main motivation was to extend the Stone Decomposition Theorem, which is the key result to prove the classical separation theorems for convex sets in the linear space setting (cf. [4], [15]). For the separation of subsemigroups, Stone type theorems were obtained by the second author in [11] and were applied for the characterization of quasideviation means in [10].
In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In the case when the underlying group is the additive group of a linear space and the endomorphisms are given as multiplication by scalars from the interval [0, 1], these notions reduce to the above mentioned concepts of convexity. The most convenient setting where our results will be formulated are Abelian groups equipped with a translation invariant metric. In this framework, we introduce the boundedness, the norm, the measure of injectivity and the spectral radius of endomorphisms motivated by the respective concepts from the theory of linear operators on normed spaces. Beyond the investigation of their properties, the first main goal of Section 2 is Theorem 2.4, which extends Rådström's celebrated cancellation theorem. Another useful result of this section is Theorem 2.6 which generalizes the Neumann Invertibility Theorem. In Section 3, we define the convexity of sets with respect to a family of endomorphisms, then we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex and we establish the basic properties of this family. One of the main results of this section is Theorem 3.6 which establishes conditions which imply the midpoint convexity. The second main result is Theorem 3.7 which, under certain regularity assumptions and using our extension of the Rådström Cancellation Theorem, presents further structural properties of the family of endomorphisms that make a subset convex.

Auxiliary concepts and tools
Let (X, +) be an Abelian group and let E(X) denote the family of all endomorphisms, i.e., additive maps T : X → X. Then, one can see that E(X) is a ring if the composition is taken as the ring multiplication. Thus, every element T ∈ E(X) generates an endomorphism T : The multiplication of the elements of X by natural numbers can be introduced via the following recursive definition: The mapping π n (x) := n·x is always an endomorphism provided that (X, +) is an Abelian group. We say that an Abelian group (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) will simply be denoted by 1 n ·x. The set of natural numbers n for which X is uniquely divisible by n will be denoted by div(X). Obviously, div(X) is a multiplicative subsemigroup of N whose unit element is 1.
For the sake of convenience, we introduce the following notation: For a subset A ⊆ X and n ∈ N, [n]A := {x 1 + · · · + x n | x 1 , . . . , x n ∈ A}.
The inclusion n·A ⊆ [n]A is obvious. In the case when the reversed inclusion is also valid we say that the set A is n-convex. For properties of n-convex sets, we refer to the paper [5]. In particular, by [5,Proposition 2], we have that if a set is n-and also m-convex, then it is (nm)-convex.
In the case when the Abelian group (X, +) is equipped with a translation invariant metric d, we say that (X, +, d) is a metric Abelian group. In a metric Abelian group, the group operations are continuous with respect to the topology induced by d. Therefore, metric groups are automatically topological groups. In a metric Abelian group (X, +, d), the d-norm · d : X → R is defined as x d := d(x, 0). From the properties of d, one can deduce that · d is a positive definite and subadditive even function on X and, for all x, y ∈ X, d(x, y) = x − y d . The positive homogeneity of the d-norm is not a consequence of the properties of a translation invariant metric. In general, the subadditivity of · d implies that n·x d ≤ n x d for all x ∈ X and n ∈ N. The equality here, however, may not be valid.
An endomorphism T : X → X is called d-bounded if there exists c ≥ 0 such that T (x) d ≤ c x d for all x ∈ X. 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. One can show that (E d (X), +, d * ) is also a metric group, where d * (T, S) := T − S * d for all T, S ∈ E d (X), furthermore, · * d is submultiplicative, which makes E d (X) a d-normed ring. 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 a d-bounded endomorphism T , we define its measure of injectivity µ d (T ) by Therefore, if µ d (T ) > 0, then T must be injective. Furthermore, in this case, with the substitution y = T (x), for the inverse map T −1 : T (X) → X, we can obtain the following inequality: If, additionally, T is surjective, then it follows that If X is the additive group of a Banach space, then the open mapping theorem implies that the inverse of bijective linear map is automatically bounded, therefore, its modulus of injectivity is positive.
For n ∈ N, the measure of injectivity of the map π n will simply be denoted by µ d (n). In this particular case we also have supermultiplicativity and a Lipschitzian property as stated below.
Proof. Let T, S ∈ E d (X) be arbitrary. Then, for x ∈ X \ {0}, the inequality (2) implies that Taking the supremum and the infimum of both sides with respect to x ∈ X \ {0}, respectively, the two first two inequalities in (4) follow. For the last inequality, observe that, for all x ∈ X \ {0}, Taking the infimum of the right hand side, it follows that interchanging the roles of S and T , we obtain the last inequality in (4).
The above lemma immediately yields the following result. Its simple proof is left to the reader.
Further properties of d-bounded maps are stated in the following lemma. Lemma 2.3. Let (X, +, d) be a metric group and T ∈ E d (X). Then T maps bounded, resp. compact subsets of X into bounded, resp. compact subsets. If µ d (T ) > 0 and either X is complete or T (X) is closed, then T maps closed subsets of X into closed subsets.
Proof. If T ∈ E d (X) then T is Lipschitzian and hence continuous (with Lipschitz modulus T * d ), therefore the first statement follows.
First assume that X is complete. By inequality (2), for all k, m ∈ N, it follows that The following result is an extension of the celebrated Rådström Cancellation Theorem (cf. [13]).
Theorem 2.4. 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 Proof. Assume that A, B, C satisfy the conditions of the theorem. First, we will prove, for all n ∈ N, by induction that For n = 1, this is just the assumption A + C ⊆ B + C. If it holds for n, then In particular, in view of the n 0 -convexity of B, for n = n k 0 , we get Let c 0 ∈ C be fixed and let a ∈ A be arbitrary. Then, by the above inclusion, for all k ∈ N, there exist b k ∈ B and c k ∈ C such that n k 0 ·a + c 0 = n k 0 ·b k + c k . Hence, applying inequality (3) with the substitutions n := n k 0 and x := a − b k together with inequality (5) of Lemma 2.1, we get By condition µ d (n 0 ) > 1, it follows that the sequence a − b k d converges to zero. Hence, the sequence (b k ) converges to a and the closedness of B now yields that a must belong to B. This proves the desired inclusion A ⊆ B.
The d-spectral radius of an endomorphism T ∈ E d (X) is defined as In view of the submultiplicativity of · * d on E d (X), for all m ∈ N, we have that Now, upon taking the upper limit as m → ∞, we get The next statement summarizes the basic properties of the spectral radius and its connection to the injectivity modulus.
Lemma 2.5. Let (X, +, d) be a metric group and let T, S ∈ E d (X) such that T • S = S • T . Then In particular, Assume that T and S are commuting endomorphisms. Then, by the binomial theorem, for all m ∈ N, we have that Let ε > 0 be arbitrary. Then, there exists n 0 ∈ N such that, for all n ≥ n 0 , Denote With this notation, for m ≥ 2n 0 − 1, from (7) and (8) we can deduce Taking the mth root of both sides and then (using elementary tools) calculating the upper limit as m → ∞, we get The number ε > 0 being arbitrary, the first inequality in (6) follows.
To prove the first part of the second inequality in (6), using Lemma 2.1, observe that Taking the m-th root and calculating the upper limit as m → ∞, we get µ d (T )ρ d (S) ≤ ρ d (T • S). The other inequality in (6) is the consequence of the inequalities Finally, putting S := I into (6), we get the last assertion.
The following result is an analogue of the so-called Neumann invertibility theorem and it can be proved almost exactly in the same way as the classical result of Carl Neumann. Theorem 2.6. 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, Proof. It easily follows from the completeness of the metric space (X, d) that E d (X) is also complete in the metric d * defined above.
From the condition ρ d (T ) < 1 it follows that there exists 0 < q < 1 such that Then, there exists an m 0 ∈ N such that, for all m ≥ m 0 , This inequality yields that the sequence S n := n k=0 T k is a Cauchy sequence in E d (X). Therefore it converges to an additive function S belonging to E d (X). One can easily see that (I − T ) • S n = S n • (I − T ) = I − T n+1 . Upon taking the limit n → ∞, we get that (I − T ) • S = S • (I − T ) = I, which proves that S is the inverse of I − T .

Corollary 2.7. Let (X, +, d) be a complete metric Abelian group and let T, S
Proof. If ρ d (T • S −1 ) < 1, then, by Theorem 2.6, I − T • S −1 is an invertible element in E d (X) and we have Therefore, S − T is also an invertible element of E d (X).
In the case when ρ d (S −1 • T ) < 1 holds, then I − S −1 • T is an invertible element of E d (X) and we get Thus, again we obtain S − T is an invertible element of E d (X).

Main results
Let (X, +) be an Abelian group. 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. [7]). 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.
The following lemma is obvious but useful. Proof. The T convexity of D is equivalent to the property that, for all p ∈ D, the inclusion holds which, by the additivity of T , is identical to the inclusion (9).
The basic properties of T -convex sets are described in the following result. Proof. The first statement is obvious. The assertions concerning intersection and chain union are easy to verify. The closedness with respect to algebraic addition is a consequence of the additivity of the elements of T.
To prove the last assertion, let A ∈ E(X) commute with any member of T and let D ∈ C T (X).

First we prove that A(D)
is T-convex. For this, let u, v ∈ A(D). Then there exist x, y ∈ D such that A(x) = u and A(y) = v. Using that D is T-convex, we have that T (x) + (I − T )(y) ∈ D for all T ∈ T. Hence, for all T ∈ T, To prove that A −1 (D) is T-convex, let x, y ∈ A −1 (D). Then A(x), A(y) ∈ D. By the T-convexity of D, for all T ∈ T, it follows that Hence, T (x) + (I − T )(y) ∈ A −1 (D) for all T ∈ T. This completes the proof of the T-convexity of A −1 (D).
In view of the closedness of C T (X) with respect to intersection, for every subset S of X, the set conv T (S) := {D ∈ C T (X) : S ⊆ D} is the smallest T-convex set containing S, which will be called the T-convex hull of S. 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 .
On the other hand, Therefore, S ∈ T D follows. This yields that T D is T -convex for all T ∈ T D , which was to be proved. The proof of the second assertion is completely analogous, the last statement follows by taking T 2 = 0 in the above argument.
In the theorem below, we show that T -convexity implies midpoint convexity under certain conditions on X and T . Proof. Let D be a nonempty T -convex subset of X. Define the sequence of operators T n by the following recursion: Then, in view of the last assertion of Corollary 3.5, we have that T n ∈ T d D for all n ∈ N. On the other hand it follows by induction that On can see that this formula is correct for n = 1. Assume that it is valid for n = k ≥ 1. Then which proves the validity of (12) for n = k + 1.
From the assumption ρ d (2 · T − I) < 1, using Lemma 2.5 we obtain that T n converges to 1 2 · I. If R, S ∈ cl T d D , then there exist sequences R n and S n in T d D converging to R and S, respectively. By Theorem 3.3, we have that T n • R n + (I − T n ) • S n ∈ T d D . Upon taking the limit, it follows that 1 2 ·(R + S) ∈ cl T d D proving that the set cl T d D is midpoint convex. To complete the proof, assume that D is a closed T -convex subset of X. Then it is also T n -convex for all n ∈ N. Then, taking the limit n → ∞, it follows that D is midpoint convex. Theorem 3.7. Let (X, +, d) be a metric Abelian group, assume that there exists n 0 ∈ N such that µ d (n 0 ) > 1. Assume that either X is complete or n 0 ·X is closed and let D be a bounded n 0 -convex set. Then, for all n ∈ N and T 1 , . . . , T n ∈ T d D , T 1 (D) + · · · + T n (D) ⊆ cl (T 1 + · · · + T n )(D) .
Proof. If D is n 0 -convex, then it is also n k 0 -convex for all k ∈ N. Let n ∈ N, T 1 , . . . , T n ∈ T d D and then choose k ∈ N so that n k 0 > n. By using the n k 0 -and T 1 -,. . . , T n -convexity of D, we arrive at an inclusion where Theorem 2.4 can be applied:

Now, taking
we can see that A + C ⊆ B + C ⊆ cl(B) + C holds. Furthermore, due to the Lipschitz property of the endomorphisms T 1 , . . . , T n and the boundedness of D, the set C is bounded. The n 0 -convexity of D easily implies that B is n 0 -convex. Using that µ d (n 0 ) > 0, Lemma 2.3 implies Thus, cl(B) is also n 0 -convex. Hence, Theorem 2.4 implies that A ⊆ cl(B), which was to be proved.
With additional properties on the set D, or on the space X, or on the operator T 1 + · · · + T n , we have the following stronger statement.
Proof. As we have seen it in the proof of Theorem 3.7, with notation (13), we have A + C ⊆ B + C. The set B is n 0 -convex. In order to deduce the inclusion A ⊆ B with the aid of Theorem 2.4, it suffices to show that in each of the cases (i), (ii), (iii), the set B is closed.
If assumption (i) is valid, then the continuity of the endomorphisms, implies that B is compact and hence it is a closed set.
If either (ii) or (iii) holds, then Lemma 2.3 with operator T := T 1 + · · · + T n implies that B = T (D) is closed. Corollary 3.9. Let (X, +, d) be a metric group, let n 0 ∈ N such that µ d (n 0 ) > 1 and let D be a compact n 0 -convex set. Then D is n-convex for all n ∈ N.
Proof. The statement immediately follows by applying the first assertion of Theorem 3.8 for the setting T 1 = · · · = T n = I.
The above theorem also implies a further closure property of the set T d D . Corollary 3.10. 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 := T 1 + · · · + T n is a bijection with T −1 ∈ E d (X). Then, for all k ∈ {1, . . . , n − 1}, we have T −1 • (T 1 + · · · + T k ) ∈ T d D . Proof. Provided that T is a bijection with d-bounded inverse, we have that assumption (iii) of Theorem 3.8 is satisfied, thus inclusion in (14) is valid. This implies that (T −1 • T 1 )(D) + · · · + (T −1 • T n )(D) ⊆ D.