Characterizations of Convex Functions of a Vector Variable via Hermite–hadamard’s Inequality

The classical Hermite-Hadamard inequality characterizes the continuous convex functions of one real variable. The aim of the present paper is to give an analogous characterization for functions of a vector variable. 1. The Hermite-Hadamard inequality In a letter sent on November 22, 1881, to the journal Mathesis (and published there two years later), Ch. Hermite [10] noted that every convex function f : [a, b] → R satisfies the inequalities f ( a + b 2 ) 1 b − a ∫ b a f (x)dx f (a) + f (b) 2 . (1) The left-hand side inequality was rediscovered ten years later by J. Hadamard [7]. Nowadays, the double inequality (1) is called the Hermite-Hadamard inequality. The interested reader can find its complete story in the historical note by D. S. Mitrinović and I. B. Lacković [12]. TheHermite-Hadamard inequality has evoked the interest ofmanymathematicians. Especially in the last three decades it has been intensively investigated and generalized in several directions. For instance, a dual Hermite-Hadamard inequality was discussed by C. P. Niculescu [13], while a complete extension of (1) to the class of n -convex functions was obtained by M. Bessenyei and Zs. Páles [1]. Likewise, M. Bessenyei and Zs. Páles [2] proved a generalization of (1) for real-valued functions defined on an open interval I ⊆ R , which are convex with respect to a so-called positive regular pair over I . For an account on various results dealing with the Hermite-Hadamard inequality, the reader is referred to the monograph by S. S. Dragomir and C. E. M. Pearce [5]. In what follows, we are concerned with Hermite-Hadamard-type inequalities for functions of a vector variable. As pointed out byC. P. Niculescu [15], the Choquet theory (see [17]) provides the framework for a natural extension of (1) to such functions. For the reader’s convenience, we briefly present here this extension. Let E be a real locally Mathematics subject classification (2000): 26B25, 26D07, 26D15.