Another refinement of the right-hand side of the Hermite-Hadamard inequality for simplices

In this paper, we establish a new refinement of the right-hand side of Hermite-Hadamard inequality for convex functions of several variables defined on simplices.

The classical Hermite-Hadamard inequality states that if f : I → R is a convex function then for all a < b ∈ I the inequality is valid. This powerful tool has found numerous applications and has been generalized in many directions (see e.g. [2] and [1]). One of those directions is its multivariate version: ). Let f : U → R be a convex function defined on a convex set U ⊂ R n and ∆ ⊂ U be an n-dimensional simplex with vertices x 0 , x 1 , . . . , x n , then is the barycenter of ∆ and the integration is with respect to the n-dimensional Lebesgue measure.
The aim of this note is to proof a refinement of the right-hand side of (1). Let us start with a set of definitions. A function f : I → R defined on an interval I is called convex if for any x, y ∈ I and t ∈ (0, 1) the inequality The barycenter of ∆ will be denoted by b. By card K we shall denote the cardinality of set K. For each k − 1-face ∆ K we calculate the average value of f over ∆ K using the formula where the integration is with respect to the k − 1-dimensional Lebesgue measure (in case k = 1 this is the counting measure). For k = 1, 2, . . . , n + 1 we define Note that the right-hand side of the inequality (1) can be rewritten as A(n + 1) ≤ A(1). It turns out, that
In the proof we shall use the following Proof of Theorem 2. We shall prove first the inequality A(n + 1) ≤ A(n). Let us use the notation from Lemma 1. For i = 0, 1, . . . , n we have (2) b Ki = 1 n n j=0 j =i Summing (2) we obtain

Now using Lemma 1 and convexity of f applied to (3) we get
thus, by the left-hand side of (1) This shows the inequality A(n + 1) ≤ A(n). The other inequalities follow by simple induction argument applying the same reasoning to all terms in A(n) etc.
Just for completeness note that similar refinement of the left-hand side of (1) can be found in [4,Corollary 2.6]. It reads as follows: Theorem 3. For a nonempty subset K of N define the simplex Σ K as follows: let A K be the affine span of ∆ K and A ′ K be the affine space of the same dimension, parallel to A K and passing through the barycenter of ∆. Then Σ K = ∆ ∩ A ′ K . For k = 1, 2, . . . , n + 1 we let Then f (b) = B(1) ≤ B(2) ≤ · · · ≤ B(n + 1) = Avg(f, ∆).