Characterizations of convex functions of a vector variable via Hermite-Hadamard's inequality

Abstract

Abstract. 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. The Hermite-Hadamard inequalityIn 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 :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]. The Hermite-Hadamard inequality has evoked the interest of many mathematicians. 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 by C. 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

Improvements of the Hermite-Hadamard inequality for the simplex

“…Improvements of the Hermite-Hadamard inequality for univariate convex functions were obtained in [11]. As for the Hermite-Hadamard inequality for multivariate convex functions, one may refer to [2, 4, 5, 12–16], and [17]. …”

On the Validity of Nonlinear and Nonsmooth Inequalities

“… is given and , to obtain the optimal solution of the NLP problem (6). We note that, for sufficiently number m , the optimal solution of the NLP problem (6) is an appro ximate optimal solution for the NLP problem (2).…”

“…Assume (see [6,7]). Here, by assumption (where M is a sufficiently big number), the infinite dimensional nonsmooth optimization problem (11) can be approximated to the follo wing finite d imensional smooth problem (see [12]): (12) Further, in [11,12] , to obtain the optimal solution of the NLP problem (15), and then obtain an approximate optimal solution for the nonsmooth optimization problem (2) to determine the validity of nonsmooth inequality (1).…”

“…Several generalizat ions and proofs are given related to the inequalities, such as Jensen type inequalities [1][2][3][4], Hermite-Hadamard inequality [5][6][7][8][9][10], etc. But, it should always be remembered that there is no standard way of proof and there is no general rule in choosing techniques to be used.…”

Multi-dimensional Hadamard's inequalities

“…We extend our . The reader is referred to [5] and [6] for the original papers on Hadamard's inequality, and [1], [4], [7], [8], [9], [10] or [13] for some new developments on this topic.…”