# Improvements of the Hermite-Hadamard inequality for the simplex

## Abstract

In this study, the simplex whose vertices are barycenters of the given simplex facets plays an essential role. The article provides an extension of the Hermite-Hadamard inequality from the simplex barycenter to any point of the inscribed simplex except its vertices. A two-sided refinement of the generalized inequality is obtained in completion of this work.

## 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 $$\mathbb{R}$$.

## 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, 1216], and [17].

## 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 $$\mathcal{B}'=\mathcal{B}\setminus\{B_{1},\ldots,B_{n+1}\}$$.

### 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].

## 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).

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## Acknowledgements

This work has been fully supported by Mechanical Engineering Faculty in Slavonski Brod, and the Croatian Science Foundation under the project HRZZ-5435. The author wishes to thank Velimir Pavić who graphically prepared Figure 1.

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Correspondence to Zlatko Pavić.

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Pavić, Z. Improvements of the Hermite-Hadamard inequality for the simplex. J Inequal Appl 2017, 3 (2017). https://doi.org/10.1186/s13660-016-1273-z

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• DOI: https://doi.org/10.1186/s13660-016-1273-z

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### Keywords

• convex combination
• simplex