1 Introduction

A concise approach to the concept of affinity and convexity is as follows. Let \(\mathbb{X}\) be a linear space over the field \(\mathbb{R}\). Let \(P_{1},\ldots,P_{m}\in\mathbb{X}\) be points, and let \(\lambda_{1},\ldots ,\lambda_{m}\in\mathbb{R}\) be coefficients. A linear combination

$$ \sum_{j=1}^{m} \lambda_{j}P_{j} $$
(1)

is affine if \(\sum_{j=1}^{m}\lambda_{j}=1\). An affine combination is convex if all coefficients \(\lambda_{j}\) are nonnegative.

Let \(\mathcal{S}\subseteq\mathbb{X}\) be a set. The set containing all affine combinations of points of \(\mathcal{S}\) is called the affine hull of the set \(\mathcal{S}\), and it is denoted with \(\operatorname{aff}\mathcal{S}\). A set \(\mathcal{S}\) is affine if \(\mathcal{S}=\operatorname{aff}\mathcal{S}\). Using the adjective convex instead of affine, and the prefix conv instead of aff, we obtain the characterization of the convex set.

A convex function \(f:\operatorname{conv}\mathcal{S}\to\mathbb{R}\) satisfies the Jensen inequality

$$ f \Biggl(\sum_{j=1}^{m} \lambda_{j}P_{j} \Biggr) \leq\sum _{j=1}^{m}\lambda_{j}f(P_{j}) $$
(2)

for all convex combinations of points \(P_{j}\in\mathcal{S}\). An affine function \(f:\operatorname{aff}\mathcal{S}\to\mathbb{R}\) satisfies the equality in equation (2) for all affine combinations of points \(P_{j}\in\mathcal{S}\).

Throughout the paper, we use the n-dimensional space \(\mathbb {X}=\mathbb{R}^{n}\) over the field \(\mathbb{R}\).

2 Convex functions on the simplex

The section is a review of the known results on the Hermite-Hadamard inequality for simplices, and it refers to its generic background. The main notification is concentrated in Lemma 2.1, which is also the generalization of the Hermite-Hadamard inequality.

Let \(A_{1},\ldots,A_{n+1}\in\mathbb{R}^{n}\) be points so that the points \(A_{1}-A_{n+1},\ldots,A_{n}-A_{n+1}\) are linearly independent. The convex hull of the points \(A_{i}\) written in the form of \(A_{1} \cdots A_{n+1}\) is called the n-simplex in \(\mathbb{R}^{n}\), and the points \(A_{i}\) are called the vertices. So, we use the denotation

$$ A_{1}\cdots A_{n+1}=\operatorname{conv} \{A_{1},\ldots,A_{n+1}\}. $$
(3)

The convex hull of n vertices is called the facet or \((n-1)\)-face of the given n-simplex.

The analytic presentation of points of an n-simplex \(\mathcal {A}=A_{1}\cdots A_{n+1}\) in \(\mathbb{R}^{n}\) arises from the n-volume by means of the Lebesgue measure or the Riemann integral. We will use the abbreviation vol instead of \(\operatorname{vol}_{n}\).

Let \(A\in\mathcal{A}\) be a point, and let \(\mathcal{A}_{i}\) be the convex hull of the set containing the point A and vertices \(A_{j}\) for \(j\neq i\), formally as

$$ \mathcal{A}_{i}=\operatorname{conv}\{A_{1}, \ldots,A_{i-1},A,A_{i+1},\ldots ,A_{n+1}\}. $$
(4)

Each \(\mathcal{A}_{i}\) is a facet or n-subsimplex of \(\mathcal{A}\), so \(\operatorname{vol}(\mathcal{A}_{i})=0\) or \(0<\operatorname{vol}(\mathcal {A}_{i})\leq\operatorname{vol}(\mathcal{A})\), respectively. The sets \(\mathcal{A}_{i}\) satisfy \(\mathcal{A}=\bigcup_{i=1}^{n+1}\mathcal{A}_{i}\) and \(\operatorname{vol}(\mathcal{A}_{i}\cap\mathcal{A}_{j})=0\) for \(i\neq j\), and so it follows that \(\operatorname{vol}(\mathcal{A})=\sum_{i=1}^{n+1}\operatorname{vol}(\mathcal{A}_{i})\).

The point A can be uniquely represented as the convex combination of the vertices \(A_{i}\) by means of

$$ A=\sum_{i=1}^{n+1} \alpha_{i}A_{i}, $$
(5)

where we have the coefficients

$$ \alpha_{i}=\frac{\operatorname{vol}(\mathcal{A}_{i})}{\operatorname{vol}(\mathcal{A})}. $$
(6)

If the point A belongs to the interior of the n-simplex \(\mathcal {A}\), then all sets \(\mathcal{A}_{i}\) are n-simplices, and consequently all coefficients \(\alpha_{i}\) are positive. Furthermore, the reverse implications are valid.

If μ is a positive measure on \(\mathbb{R}^{n}\), and if \(\mathcal {S}\subseteq\mathbb{R}^{n}\) is a measurable set such that \(\mu(\mathcal{S})>0\), then the integral mean point

$$ S= \biggl(\frac{\int_{\mathcal{S}}x_{1} \,d\mu(x)}{\mu(\mathcal {S})},\ldots, \frac{\int_{\mathcal{S}}x_{n} \,d\mu(x)}{\mu(\mathcal {S})} \biggr) $$
(7)

is called the μ-barycenter of the set \(\mathcal{S}\). In the above integrals, points \(x\in\mathcal{S}\) are used as \(x=(x_{1},\ldots ,x_{n})\). The μ-barycenter S belongs to the convex hull of \(\mathcal{S}\). When we use the Lebesgue measure, we say just barycenter. If \(\mathcal{S}\) is closed and convex, then a μ-integrable continuous convex function \(f:\mathcal{S}\to\mathbb{R}\) satisfies the inequality

$$ f \biggl(\frac{\int_{\mathcal{S}}x_{1} \,d\mu(x)}{\mu(\mathcal {S})},\ldots, \frac{\int_{\mathcal{S}}x_{n} \,d\mu(x)}{\mu(\mathcal {S})} \biggr) \leq \frac{\int_{\mathcal{S}}f(x) \,d\mu(x)}{\mu(\mathcal{S})} $$
(8)

as a special case of Jensen’s inequality for multivariate convex functions; see the excellent McShane paper in [1]. If f is affine, then the equality is valid in (8).

We consider a convex function f defined on the n-simplex \(\mathcal {A}=A_{1}\cdots A_{n+1}\). The following lemma presents a basic inequality for a convex function on the simplex, and it refers to the connection of the simplex barycenter with simplex vertices.

Lemma 2.1

Let μ be a positive measure on \(\mathbb{R}^{n}\). Let \(\mathcal {A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb{R}^{n}\) such that \(\mu(\mathcal{A})>0\). Let A be the μ-barycenter of \(\mathcal {A}\), and let \(\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be its unique convex combination by means of

$$ A= \biggl(\frac{\int_{\mathcal{A}}x_{1} \,d\mu(x)}{\mu(\mathcal {A})},\ldots, \frac{\int_{\mathcal{A}}x_{n} \,d\mu(x)}{\mu(\mathcal {A})} \biggr) = \sum_{i=1}^{n+1}\alpha_{i}A_{i}. $$
(9)

Then each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality

$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr)\leq \frac{\int_{\mathcal{A}}f(x) \,d\mu(x)}{\mu(\mathcal{A})} \leq\sum _{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
(10)

Proof

We have three cases depending on the position of the μ-barycenter A within the simplex \(\mathcal{A}\).

If A is an interior point of \(\mathcal{A}\), then we take a supporting hyperplane \(x_{n+1}=h_{1}(x)\) at the graph point \((A,f(A))\), and the secant hyperplane \(x_{n+1}=h_{2}(x)\) passing through the graph points \((A_{1},f(A_{1})),\ldots,(A_{n+1},f(A_{n+1}))\). Using the affinity of the functions \(h_{1}\) and \(h_{2}\), we get

$$\begin{aligned} f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) =& h_{1}(A) = \frac{\int_{\mathcal{A}}h_{1}(x) \,d\mu(x)}{\mu(\mathcal{A})} \\ \leq& \frac{\int_{\mathcal{A}}f(x) \,d\mu(x)}{\mu(\mathcal{A})} \\ \leq& \frac{\int_{\mathcal{A}}h_{2}(x) \,d\mu(x)}{\mu(\mathcal{A})} = h_{2}(A) \\ =& \sum_{i=1}^{n+1}\alpha_{i}h_{2}(A_{i}) = \sum_{i=1}^{n+1}\alpha_{i}f(A_{i}) \end{aligned}$$
(11)

because \(h_{2}(A_{i})=f(A_{i})\). So, formula (10) works for the interior point A.

If A is a relative interior point of a certain k-face where \(1 \leq k \leq n-1\), then we can apply the previous procedure to the respective k-simplex. For example, if \(A_{1}\cdots A_{k+1}\) is the observed k-face, then the coefficients \(\alpha_{1},\ldots,\alpha_{k+1}\) are positive, and the coefficients \(\alpha_{k+2},\ldots,\alpha_{n+1}\) are equal to zero.

If A is a simplex vertex, suppose that \(A=A_{1}\), then the trivial inequality \(f(A_{1})\leq f(A_{1})\leq f(A_{1})\) represents formula (10). □

More generally, if the μ-barycenter A lies in the interior of \(\mathcal{A}\), the inequality in formula (10) holds for all μ-integrable functions \(f:\mathcal{A}\to\mathbb{R}\) that admit a supporting hyperplane at A, and satisfy the supporting-secant hyperplane inequality

$$ h_{1}(x) \leq f(x) \leq h_{2}(x) $$
(12)

for every point x of the simplex \(\mathcal{A}\).

Lemma 2.1 was obtained in [2], Corollary 1, the case \(\alpha_{i}=1/(n+1)\) was obtained in [3], Theorem 2, and a similar result was obtained in [4], Theorem 2.4.

By applying the Lebesgue measure or the Riemann integral in Lemma 2.1, the condition in (9) gives the barycenter

$$ A= \biggl(\frac{\int_{\mathcal{A}}x_{1} \,dx}{\operatorname{vol}(\mathcal {A})},\ldots, \frac{\int_{\mathcal{A}}x_{n} \,dx}{\operatorname{vol}(\mathcal{A})} \biggr) = \frac{\sum_{i=1}^{n+1}A_{i}}{n+1}, $$
(13)

and its use in formula (10) implies the Hermite-Hadamard inequality

$$ f \biggl(\frac{\sum_{i=1}^{n+1}A_{i}}{n+1} \biggr)\leq \frac{\int_{\mathcal{A}}f(x) \,dx}{\operatorname{vol}(\mathcal{A})} \leq \frac{\sum_{i=1}^{n+1}f(A_{i})}{n+1}. $$
(14)

The above inequality was introduced by Neuman in [5]. An approach to this inequality can be found in [6].

The discrete version of Lemma 2.1 contributes to the Jensen inequality on the simplex.

Corollary 2.2

Let \(\mathcal{A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb {R}^{n}\), and let \(P_{1},\ldots,P_{m}\in\mathcal{A}\) be points. Let \(A=\sum_{j=1}^{m}\lambda_{j}P_{j}\) be a convex combination of the points \(P_{j}\), and let \(\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be the unique convex combination of the vertices \(A_{i}\) such that

$$ A=\sum_{j=1}^{m} \lambda_{j}P_{j}=\sum_{i=1}^{n+1} \alpha_{i}A_{i}. $$
(15)

Then each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality

$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq\sum _{j=1}^{m}\lambda_{j}f(P_{j}) \leq\sum_{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
(16)

Proof

The discrete measure μ concentrated at the points \(P_{j}\) by the rule

$$ \mu \bigl(\{P_{j}\} \bigr)=\lambda_{j} $$
(17)

can be utilized in Lemma 2.1 to obtain the discrete inequality in formula (16). □

Putting \(\sum_{j=1}^{m}\lambda_{j}P_{j}\) instead of \(\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) within the first term of formula (16), we obtain the Jensen inequality extended to the right.

Corollary 2.2 in the case \(\alpha_{i}=1/(n+1)\) was obtained in [3], Corollary 4.

One of the most influential results of the theory of convex functions is the Jensen inequality (see [7] and [8]), and among the most beautiful results is certainly the Hermite-Hadamard inequality (see [9] and [10]). A significant generalization of the Jensen inequality for multivariate convex functions can be found in [1]. 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].

3 Main results

Throughout the section, we will use an n-simplex \(\mathcal {A}=A_{1}\cdots A_{n+1}\) in the space \(\mathbb{R}^{n}\), and its two n-subsimplices which will be denoted with \(\mathcal{B}\) and \(\mathcal{C}\).

Let \(B_{i}\) stand for the barycenter of the facet of \(\mathcal{A}\) not containing the vertex \(A_{i}\) by

$$ B_{i}=\frac{\sum_{i \neq j=1}^{n+1}A_{j}}{n}, $$
(18)

and let \(\mathcal{B}=B_{1}\cdots B_{n+1}\) be the n-simplex of the vertices \(B_{i}\).

The simplices \(\mathcal{A}\) and \(\mathcal{B}\) in our three-dimensional space are tetrahedrons presented in Figure 1. Our aim is to extend the Hermite-Hadamard inequality to all points of the inscribed simplex \(\mathcal{B}\) excepting its vertices. So, we focus on the non-peaked simplex \(\mathcal{B}'=\mathcal{B}\setminus\{B_{1},\ldots,B_{n+1}\}\).

Figure 1
figure 1

The inscribed simplex as the barycenter extension.

Full size image

Lemma 3.1

Let \(\mathcal{A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb {R}^{n}\), and let \(A=\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be a convex combination of the vertices \(A_{i}\).

The point A belongs to the n-simplex \(\mathcal{B}=B_{1}\cdots B_{n+1}\) if and only if the coefficients \(\alpha_{i}\) satisfy \(\alpha _{i}\leq1/n\).

The point A belongs to the non-peaked simplex \(\mathcal{B}'=\mathcal {B}\setminus\{B_{1},\ldots,B_{n+1}\}\) if and only if the coefficients \(\alpha_{i}\) satisfy \(0<\alpha_{i}\leq1/n\).

Proof

The first statement, relating to the simplex \(\mathcal{B}\), will be covered as usual by proving two directions.

Let us assume that the coefficients \(\alpha_{i}\) satisfy the limitations \(\alpha_{i}\leq1/n\). Then the coefficients

$$ \beta_{i}=1-n\alpha_{i} $$
(19)

are nonnegative, and their sum is equal to 1. Since \(\beta_{i}=1-\sum_{i \neq j=1}^{n+1}\beta_{j}\), the reverse connection

$$ \alpha_{i}=\frac{\sum_{i \neq j=1}^{n+1}\beta_{j}}{n} $$
(20)

follows. The last of the convex combinations

$$\begin{aligned} \begin{aligned}[b] A &= \sum_{i=1}^{n+1} \alpha_{i}A_{i} \\ &= \sum_{i=1}^{n+1}\frac{\sum_{i \neq j=1}^{n+1}\beta_{j}}{n}A_{i} =\sum_{i=1}^{n+1}\beta_{i} \frac{\sum_{i \neq j=1}^{n+1}A_{j}}{n} \\ &= \sum_{i=1}^{n+1}\beta_{i}B_{i} \end{aligned} \end{aligned}$$
(21)

confirms that the point A belongs to the simplex \(\mathcal{B}\).

Let us assume that the point A belongs to the simplex \(\mathcal{B}\). Then we have the convex combination \(A=\sum_{i=1}^{n+1}\lambda_{i}B_{i}\). Using equation (21) in the reverse direction, we get the convex combinations equality

$$ \sum_{i=1}^{n+1} \lambda_{i}B_{i} =\sum_{i=1}^{n+1} \alpha_{i}A_{i} $$
(22)

with the coefficient connections \(\alpha_{i}=\sum_{i\neq j=1}^{n+1}\lambda_{j}/n\) from which we may conclude that \(\alpha_{i}\leq1/n\).

The second statement, relating to the non-peaked simplex \(\mathcal {B}'\), follows from the first statement and the convex combinations in formula (18) which uniquely represent the facet barycenters \(B_{i}\). □

We need another subsimplex of \(\mathcal{A}\). Let A be a point belonging to the interior of \(\mathcal{A}\). In this case, the sets \(\mathcal{A}_{i}\) defined by formula (4) are n-simplices. Let \(C_{i}\) stand for the barycenter of the simplex \(\mathcal{A}_{i}\) by means of

$$ C_{i}=\frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1}, $$
(23)

and let \(\mathcal{C}=C_{1}\cdots C_{n+1}\) be the n-simplex of the vertices \(C_{i}\).

Lemma 3.2

Let \(\mathcal{A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb {R}^{n}\), and let \(A=\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be a convex combination of the vertices \(A_{i}\) with coefficients \(\alpha _{i}\) satisfying \(\alpha_{i}>0\).

The point A belongs to the non-peaked simplex \(\mathcal{C}'=\mathcal {C}\setminus\{C_{1},\ldots,C_{n+1}\}\) if and only if the coefficients \(\alpha_{i}\) satisfy the additional limitations \(\alpha_{i}\leq1/n\).

Proof

Suppose that the coefficients \(\alpha_{i}\) satisfy \(0<\alpha_{i}\leq 1/n\). Let \(\beta_{i}\) be the coefficients as in equation (19). Using the trivial equality \(A=A/(n+1)+nA/(n+1)\), and the coefficient connections of equation (20), we get

$$\begin{aligned} A =& \sum_{i=1}^{n+1} \alpha_{i}A_{i} =\frac{1}{n+1}A +\frac{n}{n+1}\sum _{i=1}^{n+1}\alpha_{i}A_{i} \\ =& \sum_{i=1}^{n+1}\beta_{i} \frac{A}{n+1} +\sum_{i=1}^{n+1}\sum _{i \neq j=1}^{n+1}\beta_{j} \frac{A_{i}}{n+1} \\ =& \sum_{i=1}^{n+1}\beta_{i} \frac{A}{n+1} +\sum_{i=1}^{n+1} \beta_{i}\sum_{i \neq j=1}^{n+1} \frac{A_{j}}{n+1} \\ =& \sum_{i=1}^{n+1}\beta_{i} \frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} =\sum_{i=1}^{n+1} \beta_{i}C_{i} \end{aligned}$$
(24)

indicating that the point A lies in the simplex \(\mathcal{C}\). To show that the convex combination \(\sum_{i=1}^{n+1}\beta_{i}C_{i}\) does not represent any vertex, we will assume that some \(\beta_{i_{0}}=1\). Then \(\alpha_{i_{0}}=0\) as opposed to the assumption that all \(\alpha _{i}\) are positive.

The proof of the reverse implication goes exactly in the same way as in the proof of Lemma 3.1. □

Each simplex \(\mathcal{C}\) is homothetic to the simplex \(\mathcal{B}\). Namely, combining equations (23) and (18), we can represent each vertex \(C_{i}\) by the convex combination

$$ C_{i}=\frac{1}{n+1}A+\frac{n}{n+1}B_{i}. $$
(25)

Then it follows that

$$ C_{i}-A=\frac{n}{n+1}(B_{i}-A), $$

and using free vectors, we have the equality \(\overrightarrow {AC_{i}}=(n/(n+1))\overrightarrow{AB_{i}}\). So, the simplices \(\mathcal{C}\) and \(\mathcal{B}\) are similar respecting the homothety with the center at A and the coefficient \(n/(n+1)\).

If \(A\in\mathcal{B}'\), then \(\mathcal{C}\subset\mathcal{B}'\) by the convex combinations in formula (25). Combining Lemma 3.1 and Lemma 3.2, and applying Corollary 2.2 to the simplex inclusions \(\mathcal{C}\subset \mathcal{B}\) and \(\mathcal{B}\subset\mathcal{A}\), we get the Jensen type inequality as follows.

Corollary 3.3

Let \(\mathcal{A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb {R}^{n}\), let \(A=\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be a convex combination of the vertices \(A_{i}\) with coefficients \(\alpha _{i}\) satisfying \(0<\alpha_{i}\leq1/n\), and let \(\beta_{i}=1-n\alpha_{i}\).

Then it follows that

$$ \sum_{i=1}^{n+1} \beta_{i}C_{i} = \sum_{i=1}^{n+1} \beta_{i}B_{i} = \sum_{i=1}^{n+1} \alpha_{i}A_{i}, $$
(26)

and each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality

$$ \sum_{i=1}^{n+1} \beta_{i}f(C_{i}) \leq\sum_{i=1}^{n+1} \beta_{i}f(B_{i}) \leq\sum_{i=1}^{n+1} \alpha_{i}f(A_{i}). $$
(27)

The point A used in the previous corollary lies in the interior of the simplex \(\mathcal{A}\) because the coefficients \(\alpha_{i}\) are positive. In that case, the sets \(\mathcal{A}_{i}\) are n-simplices, and they will be used in the main theorem that follows.

Theorem 3.4

Let \(\mathcal{A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb {R}^{n}\), let \(A=\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be a convex combination of the vertices \(A_{i}\) with coefficients \(\alpha _{i}\) satisfying \(0<\alpha_{i}\leq1/n\), and let \(\beta_{i}=1-n\alpha_{i}\). Let \(\mathcal{A}_{i}\) be the simplices defined by formula (4).

Then each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality

$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq \sum _{i=1}^{n+1}\beta_{i} \frac{\int_{\mathcal{A}_{i}}f(x) \,dx}{ \operatorname{vol}(\mathcal{A}_{i})} \leq \sum_{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
(28)

Proof

Using the convex combinations equality \(\sum_{i=1}^{n+1}\alpha _{i}A_{i}=\sum_{i=1}^{n+1}\beta_{i}C_{i}\), and applying the Jensen inequality to \(f (\sum_{i=1}^{n+1}\beta _{i}C_{i} )\), we get

$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq \sum _{i=1}^{n+1}\beta_{i}f(C_{i}) = \sum_{i=1}^{n+1}\beta_{i}f \biggl( \frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} \biggr). $$

Summing the products of the Hermite-Hadamard inequalities for the function f on the simplices \(\mathcal{A}_{i}\) and the coefficients \(\beta_{i}\), it follows that

$$ \sum_{i=1}^{n+1} \beta_{i}f \biggl(\frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} \biggr) \leq \sum _{i=1}^{n+1}\beta_{i} \frac{\int_{\mathcal{A}_{i}}f(x) \,dx}{ \operatorname{vol}(\mathcal{A}_{i})} \leq \sum_{i=1}^{n+1}\beta_{i} \frac{f(A)+\sum_{i \neq j=1}^{n+1}f(A_{j})}{n+1}. $$

Repeating the procedure which was used for the derivation of formula (24), we obtain the series of equalities

$$\begin{aligned} \sum_{i=1}^{n+1} \beta_{i} \frac{f(A)+\sum_{i \neq j=1}^{n+1}f(A_{j})}{n+1} =& \frac{1}{n+1}f(A) + \frac{n}{n+1}\sum_{i=1}^{n+1} \beta_{i} \frac{\sum_{i \neq j=1}^{n+1}f(A_{j})}{n} \\ =& \frac{1}{n+1}f(A) +\frac{n}{n+1}\sum_{i=1}^{n+1} \frac{\sum_{i \neq j=1}^{n+ 1}\beta_{j}}{n}f(A_{i}) \\ =& \frac{1}{n+1}f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr)+\frac {n}{n+1}\sum _{i=1}^{n+1}\alpha_{i}f(A_{i}). \end{aligned}$$

Finally, applying the Jensen inequality to \(f (\sum_{i=1}^{n+1}\alpha_{i}A_{i} )\), we get the last inequality

$$ \frac{1}{n+1}f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr)+\frac {n}{n+1}\sum _{i=1}^{n+1}\alpha_{i}f(A_{i}) \leq\sum_{i=1}^{n+1}\alpha_{i}f(A_{i}). $$

Bringing together all of the above, we obtain the multiple inequality

$$\begin{aligned} f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq& \sum _{i=1}^{n+1}\beta_{i}f \biggl( \frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} \biggr) \leq \sum_{i=1}^{n+1} \beta_{i} \frac{\int_{\mathcal {A}_{i}}f(x) \,dx}{\operatorname{vol}(\mathcal{A}_{i})} \\ \leq& \sum_{i=1}^{n+1}\beta_{i} \frac{f(A)+\sum_{i \neq j=1}^{n+1}f(A_{j})}{n+1} \leq \sum_{i=1}^{n+1} \alpha_{i}f(A_{i}), \end{aligned}$$
(29)

of which the most important part is the double inequality in formula (28). □

The inequality in formula (29) is a generalization and refinement of the Hermite-Hadamard inequality. Taking the coefficients \(\alpha_{i}=1/(n+1)\), in which case \(\beta_{i}=1/(n+1)\), we realize the five terms inequality

$$\begin{aligned} f \biggl(\frac{\sum_{i=1}^{n+1}A_{i}}{n+1} \biggr) \leq& \frac{1}{n+1}\sum _{i=1}^{n+1}f \biggl(\frac{A_{i}+(n+2)\sum_{i \neq j=1}^{n+1}A_{j}}{(n+1)(n+1)} \biggr) \leq \frac{\int_{\mathcal{A}}f(x) \,dx}{\operatorname{vol}(\mathcal{A})} \\ \leq& \frac{1}{n+1}f \biggl(\frac{\sum_{i=1}^{n+1}A_{i}}{n+1} \biggr) +\frac{n}{n+1} \frac{\sum_{i=1}^{n+1}f(A_{i})}{n+1} \leq \frac{\sum_{i=1}^{n+1}f(A_{i})}{n+1}, \end{aligned}$$
(30)

where the second and fourth terms refine the Hermite-Hadamard inequality. The third term is generated from all of \(n+1\) simplices \(\mathcal {A}_{i}\). In the present case, these simplices have the same volume equal to \(\operatorname{vol}(\mathcal{A})/(n+1)\).

The inequality in formula (30) excepting the second term was obtained in [2], Theorem 2. Similar inequalities concerning the standard n-simplex were obtained in [5, 6] and [18]. Special refinements of the left and right-hand side of the Hermite-Hadamard inequality were recently obtained in [19] and [20].

4 Generalization to the function barycenter

If μ is a positive measure on \(\mathbb{R}^{n}\), if \(\mathcal {S}\subseteq\mathbb{R}^{n}\) is a measurable set, and if \(g:\mathcal{S}\to\mathbb{R}\) is a nonnegative integrable function such that \(\int_{\mathcal{S}}g(x) \,d\mu(x)>0\), then the integral mean point

$$ S= \biggl(\frac{\int_{\mathcal{S}}x_{1}g(x) \,d\mu(x)}{\int_{\mathcal {S}}g(x) \,d\mu(x)},\ldots, \frac{\int_{\mathcal{S}}x_{n}g(x) \,d\mu (x)}{\int_{\mathcal{S}}g(x) \,d\mu(x)} \biggr) $$
(31)

can be called the μ-barycenter of the function g. It is about the following measure. Introducing the measure ν as

$$ \nu(S)= \int_{\mathcal{S}}g(x) \,d\mu(x), $$
(32)

we get

$$ S= \biggl(\frac{\int_{\mathcal{S}}x_{1} \,d\nu(x)}{\nu(\mathcal {S})},\ldots, \frac{\int_{\mathcal{S}}x_{n} \,d\nu(x)}{\nu(\mathcal {S})} \biggr). $$
(33)

Thus the μ-barycenter of the function g coincides with the ν-barycenter of its domain \(\mathcal{S}\). So, the barycenter S belongs to the convex hull of the set \(\mathcal{S}\). By using the unit function \(g(x)=1\) in formula (31), it is reduced to formula (7).

Utilizing the function barycenter instead of the set barycenter, we have the following reformulation of Lemma 2.1.

Lemma 4.1

Let μ be a positive measure on \(\mathbb{R}^{n}\). Let \(\mathcal {A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb{R}^{n}\), and let \(g:\mathcal{A}\to\mathbb{R}\) be a nonnegative integrable function such that \(\int_{\mathcal{A}}g(x) \,d\mu(x)>0\). Let A be the μ-barycenter of g, and let \(\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be its unique convex combination by means of

$$ A= \biggl(\frac{\int_{\mathcal{A}}x_{1}g(x) \,d\mu(x)}{\int_{\mathcal {A}}g(x) \,d\mu(x)},\ldots, \frac{\int_{\mathcal{A}}x_{n}g(x) \,d\mu (x)}{\int_{\mathcal{A}}g(x) \,d\mu(x)} \biggr) = \sum_{i=1}^{n+1}\alpha_{i}A_{i}. $$
(34)

Then each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality

$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq \frac{\int_{\mathcal{A}}f(x)g(x) \,d\mu(x)}{ \int_{\mathcal{A}}g(x) \,d\mu(x)} \leq \sum _{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
(35)

The proof of Lemma 2.1 can be employed as the proof of Lemma 4.1 by using the measure ν in formula (32) or by utilizing the affinity of the hyperplanes \(h_{1}\) and \(h_{2}\) in the form of the equalities

$$ h_{1,2} \biggl(\frac{\int_{\mathcal{A}}x_{1}g(x) \,d\mu(x)}{\int _{\mathcal{A}}g(x) \,d\mu(x)},\ldots, \frac{\int_{\mathcal {A}}x_{n}g(x) \,d\mu(x)}{\int_{\mathcal{A}}g(x) \,d\mu(x)} \biggr) =\frac{\int_{\mathcal{A}}h_{1,2}(x)g(x) \,d\mu(x)}{ \int_{\mathcal{A}}g(x) \,d\mu(x)}. $$
(36)

Lemma 4.1 is an extension of the Fejér inequality (see [21]) to multivariable convex functions. As regards univariable convex functions, using the Lebesgue measure on \(\mathbb{R}\) and a closed interval as 1-simplex in Lemma 4.1, we get the following generalization of the Fejér inequality.

Corollary 4.2

Let \([a,b]\) be a closed interval in \(\mathbb{R}\), and let \(g:[a,b]\to \mathbb{R}\) be a nonnegative integrable function such that \(\int _{a}^{b}g(x) \,dx>0\). Let c be the barycenter of g, and let \(\alpha a+\beta b\) be its unique convex combination by means of

$$ c=\frac{\int_{a}^{b}xg(x) \,dx}{\int_{a}^{b}g(x) \,dx}=\alpha a+\beta b. $$
(37)

Then each convex function \(f:[a,b]\to\mathbb{R}\) satisfies the double inequality

$$ f(\alpha a+\beta b) \leq\frac{\int_{a}^{b}f(x)g(x) \,dx}{\int _{a}^{b}g(x) \,dx} \leq\alpha f(a)+\beta f(b). $$
(38)

Fejér used a nonnegative integrable function g that is symmetric with respect to the midpoint \(c=(a+b)/2\). Such a function satisfies \(g(x)=g(2c-x)\), and therefore

$$ \int_{a}^{b}(x-c)g(x) \,dx=0. $$

As a consequence it follows that

$$ \frac{\int_{a}^{b}xg(x) \,dx}{\int_{a}^{b}g(x) \,dx} =\frac{\int_{a}^{b}(x-c)g(x) \,dx}{\int_{a}^{b}g(x) \,dx} +\frac{\int_{a}^{b}cg(x) \,dx}{\int_{a}^{b}g(x) \,dx}= \frac{a+b}{2}, $$

and formula (38) with \(\alpha=\beta=1/2\) turns into the Fejér inequality

$$ f \biggl(\frac{a+b}{2} \biggr) \leq\frac{\int_{a}^{b}f(x)g(x) \,dx}{\int_{a}^{b}g(x) \,dx} \leq \frac{f(a)+f(b)}{2}. $$
(39)

Using the barycenters of the restrictions of g onto simplices \(\mathcal{A}_{i}\) in formula (4), we have the following generalization of Theorem 3.4.

Theorem 4.3

Let μ be a positive measure on \(\mathbb{R}^{n}\). Let \(\mathcal {A}=A_{1}\cdots A_{n+1}\) be an n-simplex in \(\mathbb{R}^{n}\), let \(A=\sum_{i=1}^{n+1}\alpha_{i}A_{i}\) be a convex combination of the vertices \(A_{i}\) with coefficients \(\alpha_{i}\) satisfying \(0<\alpha _{i}\leq1/n\), and let \(\beta_{i}=1-n\alpha_{i}\). Let \(\mathcal{A}_{i}\) be the simplices defined by formula (4), and let \(g_{i}:\mathcal{A}_{i}\to\mathbb{R}\) be nonnegative integrable functions such that \(\int_{\mathcal{A}_{i}}g_{i}(x) \,d\mu (x)>0\) and

$$ C_{i}= \biggl(\frac{\int_{\mathcal{A}_{i}}x_{1}g_{i}(x) \,d\mu(x)}{\int _{\mathcal{A}_{i}}g_{i}(x) \,d\mu(x)},\ldots, \frac{\int_{\mathcal {A}_{i}}x_{n}g_{i}(x) \,d\mu(x)}{\int_{\mathcal{A}_{i}}g_{i}(x) \,d\mu (x)} \biggr) =\frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1}. $$
(40)

Then each convex function \(f:\mathcal{A}\to\mathbb{R}\) satisfies the double inequality

$$ f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq \sum _{i=1}^{n+1}\beta_{i} \frac{\int_{\mathcal{A}_{i}}f(x)g_{i}(x) \,d\mu(x)}{ \int_{\mathcal{A}_{i}}g_{i}(x) \,d\mu(x)} \leq \sum_{i=1}^{n+1}\alpha_{i}f(A_{i}). $$
(41)

Proof

The first step of the proof is to apply Lemma 4.1 to the functions f and \(g_{i}\) on the simplex \(\mathcal{A}_{i}\) in the way of

$$ f \biggl(\frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} \biggr) \leq \frac{\int_{\mathcal{A}_{i}}f(x)g_{i}(x) \,d\mu(x)}{\int_{\mathcal {A}_{i}}g_{i}(x) \,d\mu(x)} \leq \frac{f(A)+\sum_{i \neq j=1}^{n+1}f(A_{j})}{n+1}. $$

Summing the products of the above inequalities with the coefficients \(\beta_{i}\), we obtain the double inequality that may be combined with formula (29), and so we obtain the multiple inequality

$$\begin{aligned} f \Biggl(\sum_{i=1}^{n+1} \alpha_{i}A_{i} \Biggr) \leq& \sum _{i=1}^{n+1}\beta_{i}f \biggl( \frac{A+\sum_{i \neq j=1}^{n+1}A_{j}}{n+1} \biggr) \leq \sum_{i=1}^{n+1} \beta_{i} \frac{\int_{\mathcal{A}_{i}}f(x)g_{i}(x) \,d\mu(x)}{\int_{\mathcal{A}_{i}}g_{i}(x) \,d\mu(x)} \\ \leq& \sum_{i=1}^{n+1}\beta_{i} \frac{f(A)+\sum_{i \neq j=1}^{n+1}f(A_{j})}{n+1} \leq \sum_{i=1}^{n+1} \alpha_{i}f(A_{i}) \end{aligned}$$
(42)

containing the double inequality in formula (41). □

The conditions in formula (40) require that the μ-barycenter of the function \(g_{i}\) coincides with the barycenter \(C_{i}= (A+\sum_{i \neq j=1}^{n+1}A_{j} )/(n+1)\) of the simplex \(\mathcal{A}_{i}\).

Using the Lebesgue measure and functions \(g_{i}(x)=1\), the inequality in formula (42) reduces to the inequality in formula (29).