Improvements of the Hermite-Hadamard inequality for the simplex
- Zlatko Pavić^{1}Email author
https://doi.org/10.1186/s13660-016-1273-z
© The Author(s) 2017
Received: 29 August 2016
Accepted: 6 December 2016
Published: 3 January 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.
Keywords
MSC
1 Introduction
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.
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.
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 \(\operatorname{vol}_{n}\).
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
Proof
We have three cases depending on the position of the μ-barycenter A within the simplex \(\mathcal{A}\).
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). □
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.
The discrete version of Lemma 2.1 contributes to the Jensen inequality on the simplex.
Corollary 2.2
Proof
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 \(\mathcal{C}\).
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.
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 \(B_{i}\). □
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
The proof of the reverse implication goes exactly in the same way as in the proof of Lemma 3.1. □
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 \(\beta_{i}=1-n\alpha_{i}\).
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).
Proof
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
Utilizing the function barycenter instead of the set barycenter, we have the following reformulation of Lemma 2.1.
Lemma 4.1
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
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
Proof
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).
Declarations
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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