DEMONSTRATIO MATHEMATICA Vol. XLVI No 4 2013 Flavia-Corina Mitroi, Kazimierz Nikodem and Szymon Wąsowicz HERMITE–HADAMARD INEQUALITIES FOR CONVEX SET-VALUED FUNCTIONS Abstract. The following version of the weighted Hermite–Hadamard inequalities for set-valued functions is presented: Let Y be a Banach space and F :[a, b] → cl(Y ) be a continuous set-valued function. If F is convex, then F (xμ) ⊃ 1 μ([a, b]) b  a F (x) dμ(x) ⊃ b - xμ b - a F (a)+ xμ - a b - a F (b), where μ is a Borel measure on [a, b] and xμ is the barycenter of μ on [a, b]. The converse result is also given. 1. Introduction It is well known that if a function f : I → R is convex, that is f (tx + (1 − t)y) ≤ tf (x) + (1 − t)f (y), x, y ∈ I, t ∈ [0, 1], then it satisfies the following Hermite–Hadamard double inequality (1) f  x + y 2  ≤ 1 y − x y  x f (t) dt ≤ f (x)+ f (y) 2 , x, y ∈ I, x < y. Moreover, for continuous functions f , the validity of the left or the right-hand side inequality in (1) is equivalent to the convexity of f (cf. e.g. [1], [2], [5], [6], [8]). The purpose of this note is to prove a set-valued counterpart of the weighted version of the above Hermite–Hadamard inequality. Our main theorem generalizes some earlier results of this type obtained by E. Sad- owska [11] and B. Piątek [9]. As a consequence of the theorem, we obtain a set-valued counterpart of the classical Fejér inequality. We present also a converse of the Hermite–Hadamard theorem for set-valued functions. 2000 Mathematics Subject Classification : dPrimary 26A51; Secondary 54C60, 28B20, 39B62. Key words and phrases : convex set-valued function, Hermite–Hadamard inequality.