Ohlin-Type Theorem for Convex Set-Valued Maps

A counterpart of the Ohlin theorem for convex set-valued maps is proved. An application of this result to obtain some inclusions related to convex set-valued maps in an alternative unified way is presented. In particular counterparts of the Jensen integral and discrete inequalities, the converse Jensen inequality and the Hermite–Hadamard inequalities are obtained.


Introduction
In Ohlin [12] proved the following interesting and very useful result on convex functions in a probabilistic context (as usual, E[X] denotes the expectation of the random variable X): Lemma 1 [12]. Let X 1 , X 2 be two real valued random variables such that E[X 1 ] = E[X 2 ]. If the distribution functions F X1 , F X2 cross one time, i.e. there exists t 0 ∈ R such that then for every convex function f : R → R.
For years the above Ohlin lemma was not well-known in the mathematical community. It has been rediscovered by Rajba [14], who found its various applications to the theory of functional inequalities. In [13,15,18], the Ohlin lemma is used, among others, to get a simple proof of the known Hermite-Hadamard inequalities, as well as to obtaining new Hermite-Hadamard type inequalities.
In this note we prove counterparts of the Ohlin theorem for convex setvalued maps. We present also applications of these results to obtain some inclusions connected with convex set-valued maps.

Preliminaries
Let (Y, · ) be a separable Banach space, B be the closed unit ball in Y , (Ω, A, P ) be a probability space with a non-atomic measure P and I ⊂ R be an open interval. Denote by n(Y ) the family of all nonempty subsets of Y and by cl(Y ) the family of all closed nonempty subsets of Y . For a given set-valued map G : Ω → n(Y ) the integral Ω G(ω)dP is understood in the sense of Aumann, i.e. it is the set of integrals of all integrable (in the sense of Bochner) selections of the map G (cf. [1,2]): A set-valued map G : Ω → n(Y ) is called integrable bounded if there exists a non-negative integrable function k : Ω → R such that G(ω) ⊂ k(ω)B, for all ω ∈ Ω. In this case every measurable selection of G is integrable and, consequently, the Aumann integral of G is nonempty whenever G is measurable.
The following properties of the Aumann integral will be needed in our investigations:  Recall that a set-valued map G : for all x 1 , x 2 ∈ I and t ∈ [0, 1] (see e.g. [1,3,4,8] and the references therein). Note that by (2), all values of G are convex subsets of Y .
The following lemma characterizes convex set-valued maps with values in cl(R).
Lemma 3 [8] . A set-valued map G : I → cl(R) is convex if and only if it has one of the following forms: Clearly, if G : I → cl(R) is convex and integrable bounded, then it is of the form a).

Ohlin-Type Result for Convex Set-Valued Maps
The following result is a counterpart the Ohlin lemma for convex set-valued maps.
Proof. The proof is divided into two steps. First, we assume that Y = R. Then, by Lemma 3 and the assumption that G is integrable bounded, we obtain that G is of the form G(x) = [g 1 (x), g 2 (x)], x ∈ I, where g 1 : I → R is convex and g 2 : I → R is concave. By the Ohlin lemma (Lemma 1), we have Hence, using Lemma 2(b), we get Now, assume that Y is an arbitrary separable Banach space. Take a nonzero continuous linear functional y * ∈ Y * . Since the set-valued map x → y * (G(x)), x ∈ I, is convex and has closed values in R, by the previous step, Take arbitrary z ∈ Ω G X 2 (ω) dP . By the definition of the Aumann integral, there exists an integrable selection g • X 2 of the set-valued map G • X 2 such that z = Ω g X 2 (ω) dP . Using (4), we obtain Since G is integrable bounded and the values y * G X 1 (ω) are convex, by Lemma 2(c), we get From (5) and (6), Since this condition holds for arbitrary y * ∈ Y * and, by Lemma 2(a) the set Ω G X 1 (ω) dP is convex and closed, by the separation theorem (see [16], Corollary 2.5.11), we obtain z ∈ Ω G X 1 (ω) dP and hence, using once more Lemma 2(c), Consequently, which finishes the proof.

Remark 5.
In the above proof we use the Ohlin lemma (Lemma 1) to obtain the inequalities (3). Replacing in Theorem 4 the assumptions on X 1 and X 2 (the same as in Ohlin's lemma) by any weaker conditions sufficient for (3) (for instance necessary and sufficient conditions such as in the Levin-Stečkin theorem [7]; cf. also [11]), we can obtain more general result. However, it should be emphasized that the assumptions in the Ohlin lemma are very convenient because they are simple and can be easy verified.

Applications
In this section, we present an application of the Ohlin-type lemma to obtain various inclusions related to convex set-valued maps in a simple and unified way. Some of these results (Corollaries 6-10) are known, but we present alternative proofs of them. The first result is a counterpart of the classical integral Jensen inequality.
Corollary 6 (cf. [8]). Let G : I → cl(Y ) be integrable bounded set-valued map and (Ω, A, P ) be a probability space with a non-atomic measure P . Then G is convex if and only if for every integrable random variable X : Ω → I.
If in the above corollary we take a random variable X with the distribution μ X = t 1 δ x1 + · · · + t n δ xn , where x 1 , . . . , x n ∈ I and t 1 , . . . , t n > 0 are such that t 1 + · · · + t n = 1, then we obtain a counterpart of the discrete Jensen inequality.
Proof. Take random variables X 1 , X 2 : Ω → I with the distributions Then the distribution functions F X , F Y satisfy condition (1) and Therefore, by Theorem 4, we obtain (8).
The next two corollaries are versions of the Hermite-Hadamard inequalities for convex set-valued maps.