#### Improvements of the Hermite-Hadamard inequality for the simplex

Pavic´ Journal of Inequalities and Applications
Improvements of the Hermite-Hadamard inequality for the simplex
Zlatko Pavic´
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. MSC: 26B25; 52A40
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 ., which is also the generalization of the Hermite-Hadamard inequality.
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A · · · An+ = conv{A, . . . , An+}.
The convex hull of n vertices is called the facet or (n – )-face of the given n-simplex.
The analytic presentation of points of an n-simplex A = A · · · An+ in Rn arises from
the n-volume by means of the Lebesgue measure or the Riemann integral. We will use the
abbreviation vol instead of voln.
Let A ∈ A be a point, and let Ai be the convex hull of the set containing the point A and
vertices Aj for j = i, formally as
Ai = conv{A, . . . , Ai–, A, Ai+, . . . , An+}.
A =
where we have the coefficients
Let A, . . . , An+ ∈ Rn be points so that the points A – An+, . . . , An – An+ are linearly
independent. The convex hull of the points Ai written in the form of A · · · An+ is called
the n-simplex in Rn, and the points Ai are called the vertices. So, we use the denotation
S =
S x dμ(x) , . . . , S xn dμ(x)
μ(S) μ(S)
is called the μ-barycenter of the set S. In the above integrals, points x ∈ S are used as
x = (x, . . . , xn). The μ-barycenter S belongs to the convex hull of S. When we use the
Lebesgue measure, we say just barycenter. If S is closed and convex, then a μ-integrable
continuous convex function f : S → R satisfies the inequality
S x dμ(x) , . . . , S xn dμ(x)
μ(S) μ(S)
as a special case of Jensen’s inequality for multivariate convex functions; see the excellent
McShane paper in []. If f is affine, then the equality is valid in ().
A =
A μx(Adμ) (x) , . . . , A μxn(Adμ) (x) =
Then each convex function f : A → R satisfies the double inequality
Proof We have three cases depending on the position of the μ-barycenter A within the
simplex A.
If A is an interior point of A, then we take a supporting hyperplane xn+ = h(x) at the
graph point (A, f (A)), and the secant hyperplane xn+ = h(x) passing through the graph
points (A, f (A)), . . . , (An+, f (An+)). Using the affinity of the functions h and h, we get
We consider a convex function f defined on the n-simplex A = A · · · An+. 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 . Let μ be a positive measure on Rn. Let A = A · · · An+ be an n-simplex in Rn
such that μ(A) > . Let A be the μ-barycenter of A, and let in=+ αiAi be its unique convex
combination by means of
= h(A)
because h(Ai) = f (Ai). So, formula () works for the interior point A.
If A is a relative interior point of a certain k-face where ≤ k ≤ n – , then we can
apply the previous procedure to the respective k-simplex. For example, if A · · · Ak+ is
the observed k-face, then the coefficients α, . . . , αk+ are positive, and the coefficients
αk+, . . . , αn+ are equal to zero.
If A is a simplex vertex, suppose that A = A, then the trivial inequality f (A) ≤ f (A) ≤
f (A) represents formula ().
More generally, if the μ-barycenter A lies in the interior of A, the inequality in formula
() holds for all μ-integrable functions f : A → R that admit a supporting hyperplane at
A, and satisfy the supporting-secant hyperplane inequality
h(x) ≤ f (x) ≤ h(x)
for every point x of the simplex A.
A =
Then each convex function f : A → R satisfies the double inequality
A =
nin=++Ai ,
and its use in formula () implies the Hermite-Hadamard inequality
Corollary . Let A = A · · · An+ be an n-simplex in Rn, and let P, . . . , Pm ∈ A be points.
Let A = jm= λjPj be a convex combination of the points Pj, and let in=+ αiAi be the unique
convex combination of the vertices Ai such that
Lemma . was obtained in [], Corollary , the case αi = /(n + ) was obtained in [],
Theorem , and a similar result was obtained in [], Theorem ..
By applying the Lebesgue measure or the Riemann integral in Lemma ., the condition
in () gives the barycenter
can be utilized in Lemma . to obtain the discrete inequality in formula ().
Putting jm= λjPj instead of in=+ αiAi within the first term of formula (), we obtain
the Jensen inequality extended to the right.
Corollary . in the case αi = /(n + ) was obtained in [], Corollary .
One of the most influential results of the theory of convex functions is the Jensen
inequality (see [] and []), and among the most beautiful results is certainly the
HermiteHadamard inequality (see [] and []). A significant generalization of the Jensen
inequality for multivariate convex functions can be found in []. Improvements of the
HermiteHadamard inequality for univariate convex functions were obtained in []. As for the
Hermite-Hadamard inequality for multivariate convex functions, one may refer to [, ,
, –], and [].
Figure 1 The inscribed simplex as the barycenter
extension.
3 Main results
Bi =
and let B = B · · · Bn+ be the n-simplex of the vertices Bi.
The simplices A and B in our three-dimensional space are tetrahedrons presented in
Figure . Our aim is to extend the Hermite-Hadamard inequality to all points of the
inscribed simplex B excepting its vertices. So, we focus on the non-peaked simplex B =
B \ {B, . . . , Bn+}.
follows. The last of the convex combinations
A =
Ai =
Ci =
confirms that the point A belongs to the simplex B.
Let us assume that the point A belongs to the simplex B. Then we have the convex
combination A = in=+ λiBi. Using equation () in the reverse direction, we get the convex
combinations equality
We need another subsimplex of A. Let A be a point belonging to the interior of A. In this
case, the sets Ai defined by formula () are n-simplices. Let Ci stand for the barycenter of
the simplex Ai by means of
Lemma . Let A = A · · · An+ be an n-simplex in Rn, and let A = in=+ αiAi be a convex
combination of the vertices Ai with coefficients αi satisfying αi > .
The point A belongs to the non-peaked simplex C = C \ {C, . . . , Cn+} if and only if the
coefficients αi satisfy the additional limitations αi ≤ /n.
Proof Suppose that the coefficients αi satisfy < αi ≤ /n. Let βi be the coefficients as
in equation (). Using the trivial equality A = A/(n + ) + nA/(n + ), and the coefficient
connections of equation (), we get
A =
i= i=j=
n + i=
i=j= n +
indicating that the point A lies in the simplex C. To show that the convex combination
in=+ βiCi does not represent any vertex, we will assume that some βi = . Then αi =
as opposed to the assumption that all αi are positive.
The proof of the reverse implication goes exactly in the same way as in the proof of
Lemma ..
Each simplex C is homothetic to the simplex B. Namely, combining equations () and
(), we can represent each vertex Ci by the convex combination
Then it follows that
Ci = n + A + n n+ Bi.
The point A used in the previous corollary lies in the interior of the simplex A because
the coefficients αi are positive. In that case, the sets Ai are n-simplices, and they will be
used in the main theorem that follows.
Theorem . Let A = A · · · An+ be an n-simplex in Rn, let A = in=+ αiAi be a convex
combination of the vertices Ai with coefficients αi satisfying < αi ≤ /n, and let βi = – nαi.
Let Ai be the simplices defined by formula ().
and each convex function f : A → R satisfies the double inequality
Then each convex function f : A → R satisfies the double inequality
Proof Using the convex combinations equality
Jensen inequality to f ( in=+ βiCi), we get
Summing the products of the Hermite-Hadamard inequalities for the function f on the
simplices Ai and the coefficients βi, it follows that
Repeating the procedure which was used for the derivation of formula (), we obtain the
series of equalities
of which the most important part is the double inequality in formula ().
The inequality in formula () is a generalization and refinement of the
HermiteHadamard inequality. Taking the coefficients αi = /(n + ), in which case βi = /(n + ),
n + i=
n + i=
n + i=
n + i=
Bringing together all of the above, we obtain the multiple inequality
can be called the μ-barycenter of the function g. It is about the following measure.
Introducing the measure ν as
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 + simplices Ai. In the present case, these simplices have
the same volume equal to vol(A)/(n + ).
The inequality in formula () excepting the second term was obtained in [],
Theorem . Similar inequalities concerning the standard n-simplex were obtained in [, ] and
[]. Special refinements of the left and right-hand side of the Hermite-Hadamard
inequality were recently obtained in [] and [].
4 Generalization to the function barycenter
If μ is a positive measure on Rn, if S ⊆ Rn is a measurable set, and if g : S → R is a
nonnegative integrable function such that S g(x) dμ(x) > , then the integral mean point
S =
S xg(x) dμ(x) , . . . , S xng(x) dμ(x)
S g(x) dμ(x) S g(x) dμ(x)
S =
Thus the μ-barycenter of the function g coincides with the ν-barycenter of its domain S .
So, the barycenter S belongs to the convex hull of the set S . By using the unit function
g(x) = in formula (), it is reduced to formula ().
Utilizing the function barycenter instead of the set barycenter, we have the following
reformulation of Lemma ..
Lemma . Let μ be a positive measure on Rn. Let A = A · · · An+ be an n-simplex in Rn,
and let g : A → R be a nonnegative integrable function such that A g(x) dμ(x) > . Let A
be the μ-barycenter of g, and let in=+ αiAi be its unique convex combination by means
of
A =
AAxgg(x(x))ddμμ(x(x)) , . . . , AAxgng(x(x))ddμμ(x(x)) =
Then each convex function f : A → R satisfies the double inequality
The proof of Lemma . can be employed as the proof of Lemma . by using the
measure ν in formula () or by utilizing the affinity of the hyperplanes h and h in the form
of the equalities
A xg(x) dμ(x) , . . . , A xng(x) dμ(x)
A g(x) dμ(x) A g(x) dμ(x)
Lemma . is an extension of the Fejér inequality (see []) to multivariable convex
functions. As regards univariable convex functions, using the Lebesgue measure on R and a
closed interval as -simplex in Lemma ., we get the following generalization of the Fejér
inequality.
Corollary . Let [a, b] be a closed interval in R, and let g : [a, b] → R be a nonnegative
integrable function such that ab g(x) dx > . Let c be the barycenter of g, and let αa + βb be
its unique convex combination by means of
Then each convex function f : [a, b] → R satisfies the double inequality
Fejér used a nonnegative integrable function g that is symmetric with respect to the
midpoint c = (a + b)/. Such a function satisfies g(x) = g(c – x), and therefore
(x – c)g(x) dx = .
As a consequence it follows that
Using the barycenters of the restrictions of g onto simplices Ai in formula (), we have
the following generalization of Theorem ..
let A =
Then each convex function f : A → R satisfies the double inequality
Proof The first step of the proof is to apply Lemma . to the functions f and gi on the
simplex Ai in the way of
in=+j= f (Aj)
Ci =
in=+j= f (Aj)
containing the double inequality in formula ().
with the barycenter Ci = (A +
duces to the inequality in formula ().
Competing interests
The author declares that he has no competing interests.
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 Pavic´ who graphically prepared Figure 1.
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