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# Improvements of the Hermite-Hadamard inequality for the simplex

*Journal of Inequalities and Applications*
**volume 2017**, Article number: 3 (2017)

## 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

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

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

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

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

where we have the coefficients

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

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

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*

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

### 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

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

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

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

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*

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

### Proof

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

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].

## 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

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

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

follows. The last of the convex combinations

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

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

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

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

Then it follows that

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*

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

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*

### 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

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

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

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

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

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

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

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

we get

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*

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

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

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*

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

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

As a consequence it follows that

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

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*

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

### 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

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

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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### Cite this article

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

### MSC

- 26B25
- 52A40

### Keywords

- convex combination
- simplex
- the Hermite-Hadamard inequality