Applied Mathematics and Computation 243 (2014) 74–82

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Symmetrization procedures and convexity in centrallysymmetric polytopes

http://dx.doi.org/10.1016/j.amc.2014.05.0820096-3003/� 2014 Elsevier Inc. All rights reserved.

E-mail address: allal.guessab@univ-pau.fr

Allal GuessabLaboratoire de Mathématiques et de leurs Applications, UMR CNRS 4152, Université de Pau et des Pays de l’Adour, 64000 Pau, France

a r t i c l e i n f o

This paper is dedicated to the memory ofProfessor Qazi Ibadur Rahman.

Keywords:Centrally symmetric polytopesCentral symmetrizationConvex functionsHermite–Hadamard inequalitySymmetrized functions

a b s t r a c t

Univariate symmetrization technique has many good properties. In this paper, we adoptthe high-dimensional viewpoint, and propose a new symmetrization procedure in arbitrary(convex) polytopes of Rd with central symmetry. Moreover, the obtained results are used toextend to the arbitrary centrally symmetric polytopes the well-known Hermite–Hadamardinequality for convex functions.

� 2014 Elsevier Inc. All rights reserved.

1. Motivation and aims

To motivate our geometric idea of symmetrization of functions in convex polytopes with central symmetry, we start bydescribing briefly the univariate procedure, since its simplicity allows us to analyze all the necessary steps through very sim-ple computation. In the univariate case, say on an interval ½a; b�, a very explicit simple way of symmetrization of a given realfunction f : ½a; b� ! R is to map the non-symmetric function f into a symmetric function ~f : ½a; b� ! R, which is defined by

~f ðxÞ :¼ f ðxÞ þ f ðaþ b� xÞ2

; ð8x 2 a; b½ �Þ: ð1Þ

With this univariate symmetrization technique, we can identify many good properties, that we have for the symmetrizedfunction ~f :

1. The domain ½a; b� of f and ~f is centrally symmetric with respect to the point c :¼ aþb2 , which means that if x 2 ½a; b�, then

2c � x 2 ½a; b�;2. The function ~f is symmetric with respect to the straight line which contains the point ððaþ bÞ=2;0Þ, and is perpendicular

to the x-axis in the sense that for all x on a; b½ �, it holds

~f ð2c � xÞ ¼ ~f ðxÞ;

3. The integrals of f and ~f over ½a; b� are equal;4. If moreover f is convex, then it holds for all x in ½a; b�:

f ðcÞ ¼ mint2 a;b½ �

~f ðtÞ 6 ~f ðxÞ 6 maxt2 a;b½ �

~f ðtÞ ¼ f ðaÞ þ f ðbÞ2

; ð2Þ

5. Under the convexity assumption on f, the integral of ~f satisfies the estimates:

Fig. 1.referen

A. Guessab / Applied Mathematics and Computation 243 (2014) 74–82 75

f ðcÞ ¼ ~f ðcÞ 6 1b� a

Z b

a

~f ðxÞdx 6~f ðaÞ þ ~f ðbÞ

2¼ f ðaÞ þ f ðbÞ

2; ð3Þ

which means that ~f verifies the classical double Hermite–Hadamard inequality.

Fig. 1 show graphs of a function and its symmetrized function. Coincidentally, the last inequality (3), being a trivial con-sequence of (2), was recently shown by Dragomir [2], and also by El Farissi et al. [3]. We would like to emphasize that thissymmetrized function, first proposed in [6], has been extensively studied in the context of convex functions.

It is therefore natural to try and extend this simple symmetrization approach to general multivariate centrally symmetric(convex) polytopes � Rd. To symmetrize a function of several variables, we have to assume that its domain has a certain suit-able symmetry. To make things more precise let us first introduce some terminology. We will use boldface letters to denotevectors in Rd. Throughout X will always denote a (convex) polytope with a non-empty interior (that is, the convex hull ofðnþ 1Þ vertices v0;v1; . . . ;vnf g in Rd). We will always assume that X is centrally symmetric with respect to a pointc 2 Rd. This means that for each x 2 X, the reflection of x about c; 2c � x, is also in X. We say that X has the center of sym-metry c. There is a big difference between one and higher dimension. Indeed, in one dimension, every real interval can beconsidered as a centrally symmetric interval with respect to its midpoint. That is not the case in the multivariate setting.Complete details on centrally symmetric convex polytope can be found in Ziegler’s home page and in his papers and book[13, Section 7.3]. The standard examples of such polytopes are zonotopes. Traditionally, a zonotope is defined as the image ofa cube by an affine transformation from Rn to Rd, or as a set Z such that

Z :¼ x 2 Rd : x ¼ c þXp

i¼0

aigi; �1 6 ai 6 1

( ); ð4Þ

where c; g0; . . . ; gp are vectors of Rd. In two-dimensional case all zonotopes are centrally symmetric polygons and vice versa.More generally, centrally symmetric polygons, parallelepipeds, hyper-rectangles, and cross-polytopes belong to the richclass of zonotopes. Note that any zonotope Z as defined in (4) is always centrally symmetric with respect to the pointc 2 Rd. Even more complicated centrally symmetric polytopes are the Hanner polytopes, for details on this subject we referthe reader, for instance, to [9,10].

For a given set X � Rd; Xj j and convðXÞ denote the d-dimensional Lebesgue measure and the convex hull of X, respectively.Here and in the sequel, the symbol cgðCÞ denotes the center of gravity of a set C � Rd, which is defined as

cgðCÞ :¼R

C xdxRC dx

;

assuming C is bounded and has a non-empty interior. The vertex centroid of a polytope C � Rd with vertices c0; c1; . . . ; cnf g isdefined as the average of the vertices in C:

vcðCÞ ¼ 1nþ 1

Xn

i¼0

ci:

The graphs of f ðxÞ ¼ 2xþ 3x2 � x4 þ 10 (given in blue) and its symmetrized function ~f ðxÞ ¼ 3x2 � x4 þ 10 (given in red). (For interpretation of theces to colour in this figure legend, the reader is referred to the web version of this article.)

76 A. Guessab / Applied Mathematics and Computation 243 (2014) 74–82

We recall that cgðCÞ and vcðCÞ are always located inside C, and, in general, they do not necessarily coincide.One purpose of the present contribution is to adopt the high-dimensional viewpoint, and to propose symmetrization

techniques in arbitrary polytopes of Rd with central symmetry, which will also be shown to meet all the desirable propertiesmentioned above for univariate symmetrized function (1).

The present paper is organized as follows: in Section 2 we shortly review the fundamental properties of centrally sym-metric polytopes. It turns out that the center of symmetry of any centrally symmetric polytope coincides with both its centerof gravity and vertex centroid. The purpose of Section 3 is to investigate some properties of our symmetrization procedure. Itturns out that it has all the desirable properties for univariate symmetrized function (1). Moreover, the obtained results areused to extend to the arbitrary centrally symmetric polytope the well-known Hermite–Hadamard inequality for convexfunctions. Section 4 focuses on an important subclass of centrally symmetric polytopes X � Rd that are the images undercertain affine transformations of the unit cube. In this context, we provide an explicit upper bound of the integral meanof any convex function over such domains. Finally, in Section 5, we extend our results to the class of Wright-convex func-tions. The main property of this class is that every Wright-convex function may be decomposed as a sum of a convex functionand an additive function.

2. Preliminary facts for multivariate central symmetry

We start by collecting some terminology and basic facts about centrally symmetric polytopes. Polytopes are typically notcentrally symmetric, but when they are with respect to a point c then their center of gravity and vertex centroid are preciselythe center of symmetry c. Indeed, among the many formal properties shared with the remarkable class of centrally symmet-ric polytopes, a noteworthy fact is that:

Proposition 2.1. If the polytope X is centrally symmetric with respect to c, then

cgðXÞ ¼ vcðXÞ ¼ c: ð5Þ

The proof of Proposition 2.1 will be given after the proof of Proposition 2.3.

As seen by the following example, however, the converse is false.

Remark 2.2. The ‘‘central symmetry’’ condition is essential, it cannot be removed from Proposition 2.1. Indeed, the inverse isnot true, an immediate counterexample is a simplex. It is well-known that the center of gravity and vertex centroid of anysimplex coincide, however the simplex is not centrally symmetric since each vertex is opposite a cell rather than anothervertex and each edge is opposite a face.

Throughout this paper, we denote by rc the self-inverse mapping rc : X ! Rd; rcðxÞ :¼ 2c � x. The reader may checkthat rc is a well-defined isometric involutive affine transformation that has exactly one fixed point, which is the center ofsymmetry c, and obviously satisfies the inclusion property rcðXÞ � X. In fact, we may say more about rc . Let us start bythe following observations, which show how X; rcðXÞ, VðXÞ and VðrcðXÞÞ are related. Here, for any polytope X, we have usedthe notation VðXÞ to denote the set of vertices of X.

Proposition 2.3. The central symmetry of the polytope X implies that the mapping rc satisfies the identities:

rcðXÞ ¼ X; ð6Þ

rcðVÞ ¼ VðXÞ; ð7Þ

rcðXÞj j ¼ Xj j; ð8Þ

vcðrcðXÞÞ :¼ 1nþ 1

Xn

i¼0

rcðv iÞ ¼ vcðXÞ: ð9Þ

Proof. We start with identity (6), which is simple to prove. Indeed, it is easy to check that rc is an isometry with inverse rc .Then by the central symmetry of X, the mapping rc is clearly a bijection from X to X. Hence, the transformation rc carries Xinto itself. To prove the second equality, let us pick an element y in rcðVÞ. Then, there exists a vertex v 2 VðXÞ such that

y ¼ 2c � v : ð10Þ

Assume to the contrary that y is not a vertex of X. Since, by the first identity, y also belongs to X, we can write

y ¼Xn

i¼0

tiv i; ð11Þ

A. Guessab / Applied Mathematics and Computation 243 (2014) 74–82 77

with non-negative coefficients with unit sum: ti P 0;Pn

i¼0ti ¼ 1. Since by assumption y is not one of the v i, none of ti isequal to 1. Clearly, using (10) and (11) we see that the vertex v may represented as:

v ¼ 2c � y ð12Þ

¼Xn

i¼0

tið2c � v iÞ; ð13Þ

which is a convex combination of points in X, with two coefficients different from 0 and 1. This contradicts the fact that v is avertex of X. Hence, we have shown that

rcðVÞ � VðXÞ: ð14Þ

This means that the ‘reflected’ points rcðvÞ; v 2 VðXÞ are among the vertices of X. To prove the reverse inclusionVðXÞ � rcðVÞ, it suffices to use the fact rc is an involution, then any vertex v of X can be written as v ¼ rcðv 0Þ, withv 0 ¼ rcðvÞ, which by (14) belongs to VðXÞ. Hence, rcðVÞ is exactly the set of vertices of X. It may be worth while to pointout here that the set rcðVÞ may be just a re-arrangement of VðXÞ. Identity (8) is immediate, since isometries preserve vol-ume. Finally, since rc is a bijection and by (7) it sends any vertex of X to a vertex of X, the last identity holds trivially. Thiscompletes the proof of Proposition 2.3. h

With this proposition in hand, it is now easy to prove Proposition 2.1.

Proof of Proposition 2.1. Since, for all i ¼ 0; . . . ;n,

rcðv iÞ ¼ 2c � v i;

then, with the help of identity (9) it is now easy to establish that

vcðXÞ ¼ c: ð15Þ

Next, we prove that the point c is also the center of gravity of X. As, by (6), we have

ZX

xdx ¼Z

rc ðXÞydy;

then change of variables y ¼ 2c � x, and since rc preserves the volume (this is a bijection with a constant Jacobian determi-nant equal to one), we obviously have

cgðXÞ ¼ 2c � cgðXÞ;

then clearly the claimed equality (5) is satisfied. This completes the proof of Proposition 2.1.The following result shows the invariance of central symmetry under affine transformations.

Proposition 2.4. Every image of a centrally symmetric polytope under an affine transformation is a centrally symmetric polytope.Moreover, the center of symmetry is preserved under affine transformations.

Proof. Recall that an affine transformation of Rd is a map of the form FðxÞ ¼ AðxÞ þ b, where b is some fixed vector of Rd andA is a linear transformation of Rd. Suppose we have a centrally symmetric polytope X with center of symmetry c. It is easy toverify that the image of a polytope under an affine transformation is again a polytope. Now, let y be any point in FðXÞ, thenthere exists x 2 X such that y ¼ FðxÞ. It follows immediately that

2FðcÞ � y ¼ 2FðcÞ � FðxÞ ¼ 2AðcÞ � AðxÞ þ b ¼ Að2c � xÞ þ b ¼ Fð2c � xÞ:

Since, by central symmetry of X, the point 2c � x 2 X, then the above equality shows that for all y 2 FðXÞ the point 2FðcÞ � yis also on FðXÞ. Hence, FðXÞ is a centrally symmetric polytope with center of symmetry FðcÞ, as required. h

For later use, we end this section with a technical proposition, which is obvious from the geometric point of view.

Proposition 2.5. If X � Rd is a centrally symmetric polytope with center of symmetry c, then, for any real numbers a < b, the set~X :¼ X� a; b½ � is a centrally symmetric polytope in Rdþ1 with center of symmetry c; aþb

2

� �2 Rdþ1.

Propositions 2.4 and 2.5 imply, in particular, that parallelepipeds, hyper-rectangles, as well as cross-polytopes are triviallycentrally symmetric. With help of these two Propositions, we may also construct more complicated centrally symmetricpolytopes. For instance, twisted prism. Let P � Rd be a polytope and Q :¼ P � 1f g its embedding in Rdþ1. Then, the twistedprism tpðPÞ :¼ convðQ [ �QÞ is centrally symmetric by construction. All of these observations were made by Kurt Mahler inthe 1930’s, in connection to transference principles for linear forms. See [1] for a brief historical account.

78 A. Guessab / Applied Mathematics and Computation 243 (2014) 74–82

3. Symmetrization in centrally symmetric polytopes

The core of this section is to propose our symmetrization procedure and show that it has some desired properties. Moti-vated by the univariate symmetrization technique outlined before, we shall show here that this method of symmetrizationcan also be established for arbitrary polytopes of Rd with central symmetry. We continue to assume that X is a given cen-trally symmetric polytope with non-empty interior and which is the convex hull of ðnþ 1Þ vertices v0;v1; . . . ;vnf g in Rd.Recall that due to Proposition 2.1 its center of symmetry must coincide with its center of gravity and its centroid. Hence,for the rest of the paper in all our lower bounds, we may replace 1

nþ1

Pni¼0v i by c.

We will use the following symmetrization technique.

Definition 3.1. The symmetrization of an arbitrary continuous function f : X ! R is

~f ðxÞ ¼ f ðxÞ þ f ð2c � xÞ2

: ð16Þ

We note that the symmetrized function (16) is correctly defined in our definition since by central symmetry of X the map rc

sends X to X. It is straightforward to also show that ~f is bounded from below by minx2Xf ðxÞ and from above by maxx2Xf ðxÞ.

It should be remarked in passing that if X ¼ a; b½ � � R; then the symmetrized function (1) is an obvious special case of theidentity (16) in the one-dimensional case. Hence, our proposed method of symmetrization can be viewed as a multivariategeneralization of the approach in the univariate case, recently used by Dragomir [2] and also by El Farissi et al. [3] in theirderivation of (1).

We define the notion of (generalized) barycentric coordinates in the remainder of this paper as follows: let x be an arbi-trary point of X. We call barycentric coordinates of x with respect to VðXÞ any set of real coefficients kiðxÞf gn

i¼0 depending onthe vertices of X and on x such that all the three following properties hold true:

kiðxÞP 0; i ¼ 0; . . . ; n; ð17Þ

Xn

i¼0

kiðxÞ ¼ 1; ð18Þ

x ¼Xn

i¼0

kiðxÞv i: ð19Þ

Recall that these coordinates exist for more general types of polytopes. The first result on their existence was due to Kalman[11, Theorem 2] (1961). Eq. (19) is said to be a barycentric coordinate representation of any x 2 X. The barycentric coordi-nates will be important to us in two ways:

(1) To extend to the arbitrary centrally symmetric polytope the well-known Hermite–Hadamard inequality for symme-trized convex functions.

(2) To give us an upper bound of any symmetrized convex function on arbitrary centrally symmetric polytope.

We now come back to the motivation of this symmetrization technique, which is to establish upper and lower bounds formultivariate integrals of convex functions. First, we establish some basic properties of the function ~f . We start by showingthat the symmetrized function of any arbitrary affine function has the following simple representations:Lemma 3.2. The symmetrized function ~f of any affine function, f : X! R, satisfies, for all x 2 X,

~f xð Þ ¼ ~f1

nþ 1

Xn

i¼0

v i

!; ð20Þ

¼Xn

i¼0

ki xð Þ~f ðv iÞ: ð21Þ

Proof. We first define the operator L in CðXÞ the space set of real continuous functions on X by the formula Lðf Þ ¼ ~f . It iseasy to verify that constant functions are fixed points for the operator L, in the sense that if f is a constant function thenLðf Þ ¼ f . This shows that equalities (20) and (21) are satisfied for all constant functions. Now, since L is a linear operatorfrom CðXÞ into CðXÞ, it remains to show that (20) and (21) hold for the projection functions ej; j ¼ 1; . . . ; d, defined by:

ej : x ¼ ðx1; . . . ; xdÞ 2 X # xj:

For any function ej, we clearly have by definition:

~ej xð Þ :¼ ejðxÞ þ ejð2c � xÞ2

¼ cj; j ¼ 1; . . . ;d: ð22Þ

A. Guessab / Applied Mathematics and Computation 243 (2014) 74–82 79

where cj is the jth component of c. Then, by the representation of centroid, given in Proposition 2.1, the first identity (20) isobtained by taking x ¼ 1

nþ1

Pni¼0v i in (22). For the second identity, it suffices to apply again (22) and the fact thatPn

i¼0kiðxÞ ¼ 1, to immediately deduce that, for all j ¼ 1; . . . ; d,

Xn

i¼0

ki xð Þ~ejðv iÞ ¼Xn

i¼0

ki xð Þcj ð23Þ

¼ cj: ð24Þ

Hence, equalities (20) and (21) hold for any x 2 X, as required. This completes the proof of Lemma 3.2. h

Remark 3.3. As observed by one of referee, an alternative simple proof of Eq. (20) can be stated as follows. Let us denote the stan-dard inner product in Rd by :; :h i. Since f is affine, it can be represented in the form f ðxÞ ¼ a; xh i þ b, for x 2 X and somea 2 R; b 2 R. So, we immediately get

~f ðxÞ ¼ a; xh i þ bþ a;2c � xh i þ b2

¼ a; ch i þ b ¼ f ðcÞ ¼ ~f ðcÞ:

The next result says that, as in the one dimensional case, the symmetrization procedure of Definition 3.1 assigns to any

continuous function f a unique symmetrized function ~f of equal integral.

Lemma 3.4. The symmetrized function ~f of any continuous function, f : X ! R, satisfies

ZX

~f ðxÞdx ¼Z

Xf ðxÞdx: ð25Þ

Proof. Straightforward computation. Indeed, by definition we have

ZX

~f ðxÞdx ¼ 12

ZX

f ðxÞdxþZ

Xf ð2c � xÞdx

� �:

The result now follows by applying the change of variable formula y ¼ 2c � x in the second integral. h

The following lower and upper bounds for the symmetrized function (16) will be useful throughout the paper.

Theorem 3.5. The symmetrized function ~f of any convex function, f : X! R, satisfies for all x in X the following double inequality

~f1

nþ 1

Xn

i¼0

v i

!6

~f ðxÞ 6Xn

i¼0

kiðxÞ~f ðv iÞ: ð26Þ

Equality is attained for all affine functions.

Proof. First recall that Proposition 2.1 says that the point c is also the centroid of X. So we can start from Proposition 2.1 andobtain

1nþ 1

Xn

i¼0

v i ¼ c ¼ xþ ð2c � xÞ2

; ð8x 2 XÞ: ð27Þ

Therefore, by virtue of the Jensen’s inequality of f, we get the left-hand side of (26). For the right hand side, let us pick anyx 2 X. This point has a representation with barycentric coordinates:

x ¼Xn

i¼0

kiðxÞv i:

Since both f and f � rc are convex, then exploiting this barycentric coordinate representation and the Jensen’s inequalityonce more we derive

~f ðxÞ ¼ f ðxÞ þ f � rcðxÞ2

612

Xn

i¼0

kiðxÞðf ðv iÞ þ f � rcðv iÞÞ ¼Xn

i¼0

kiðxÞ~f ðv iÞ:

This shows that the double inequality (26) holds. Finally, by Lemma 3.2, the case of equality is easily verified. This completesthe proof of Theorem 3.5. h

Theorem 3.5 is a key ingredient in establishing the proof of the next theorem, which tells us that the symmetrized func-tion of any convex function satisfies necessary the Hermite–Hadamard (double) inequality:

80 A. Guessab / Applied Mathematics and Computation 243 (2014) 74–82

Theorem 3.6. Let l be a probability measure on X. Then, the symmetrized function ~f of any convex function f : X! R satisfies thefollowing double inequality

~f1

nþ 1

Xn

i¼0

v i

!6

ZX

~f ðxÞ dlðxÞ 6Xn

i¼0

xi~f ðv iÞ; ð28Þ

where xi ¼R

X kiðxÞdlðxÞ; i ¼ 0; . . . ;n. Equality is attained for all affine functions.

Proof. Just multiply both sides of (26) by dlðxÞ and integrate the resulting inequalities with respect to x. Finally, Lemma 3.2immediately implies the case of equality. h

In order to justify our next considerations, let us start with an easy two dimensional example showing the somewhat sur-prising fact that a non-convex function may have a convex symmetrized function. We take the rectangle X of verticesv0 ¼ ð�1;0Þ; v1 ¼ ð1;0Þ; v2 ¼ ð1;1Þ and v3 ¼ ð�1;1Þ. A simple calculation shows that ~f ðx; yÞ ¼ 3=2 is the symmetrizedfunction associated to the function f ðx; yÞ ¼ x3 þ yþ 1. Then, it is easy to see that f is clearly not convex on X, while its sym-metrized function is.

Theorem 3.7. Let l be a probability measure on X. If the symmetrized function ~f of f is convex, then the following doubleinequality holds

~fZ

XxdlðxÞ

� �6

ZX

~f ðxÞ dlðxÞ 6Xn

i¼0

xi~f ðv iÞ; ð29Þ

where xi ¼R

X kiðxÞdlðxÞ; i ¼ 0; . . . ;n. Equality is attained for all affine functions.

Proof. The left hand inequality in (29) is a simple consequence of the well known integral Jensen’s inequality. For the righthand side, we use the Jensen’s inequality of ~f to get, for all x in X,

~f ðxÞ ¼ ~fXn

i¼0

kiðxÞv i

!6

Xn

i¼0

kiðxÞ~f ðv iÞ:

Multiplying the above inequality with l and integrating over X yields

ZX

~f ðxÞdlðxÞ 6Xn

i¼0

ZX

kiðxÞdlðxÞ� �

~f ðv iÞ: ð30Þ

Hence, the double inequality (29) is established and the proof is complete. h

As a corollary, any continuous function (not necessary convex), which has a convex symmetrized function satisfies thefollowing Hermite–Hadamard type inequality.

Corollary 3.8. If the symmetrized function ~f of f is convex, then the following double inequality holds

~f1

nþ 1

Xn

i¼0

v i

!¼ f

1nþ 1

Xn

i¼0

v i

!6

ZX

f ðxÞ dx 6Xn

i¼0

xi~f ðv iÞ; ð31Þ

where xi ¼R

X kiðxÞdx; i ¼ 0; . . . ;n. Equality is attained for all affine functions.

Proof. The assertion follows directly from Theorem 3.7. Indeed, since X is a centrally symmetric polytope, then the center ofgravity 1

Xj jR

X xdx coincides with the vertex centroid 1nþ1

Pni¼0v i. Hence, the conclusion now follows from Theorem 3.7 and

Lemma 3.4. h

4. Affine images

Since it appears that no explicit formula for barycentric coordinates for more general types of centrally symmetricpolytopes is known, then, the upper bounds given in Theorems 3.6 and 3.7 are of limited use when one thinks to applicationsin numerical analysis, see [5,7]. This section focuses on an important subclass of centrally symmetric polytopes X � Rd thatare images under certain affine transformations F : Rd ! Rd, i. e., X ¼ FðbXÞ, where bX :¼ �1;1½ �d denotes the d-dimensionalunit cube. Of course this is under the assumption that F is a bijection. Particular examples of the large variety of suchpolytopes are centrally symmetric polygons, parallelepipeds, hyper-rectangles. Furthermore, we will consider only the

A. Guessab / Applied Mathematics and Computation 243 (2014) 74–82 81

unweighted (or ‘‘weight dlðxÞ ¼ dx’’) case. This approach will allow us to determine a simple explicit form of the coefficientsxi ¼

RX kiðxÞdx; i ¼ 0; . . . ;n, in Theorems 3.6 and 3.7.

We want to infer some consequences concerning the images under affine transformations. Since barycentric coordinatesdo not change under affine transformation, then the coefficients xi; i ¼ 0; . . . ;n, are given as

xi ¼ Xj j bxi; ð32Þ

where, bxi ¼RbX bkiðbxÞdbx; i ¼ 0; . . . ;n with bki; i ¼ 0; . . . ;n being the barycentric coordinates with respect to the vertices of bX.

Due to invariance under permutation of the variables on the unit cube, we can see that the integrals of the barycentriccoordinates are equal, and since

Pni¼0bkiðbxÞ ¼ 1, we immediately deduce that

xi ¼Xj j2d

; i ¼ 0; . . . ;n: ð33Þ

Thus, we are able now to prove the following Theorems, which summarize the discussion above:

Theorem 4.1. If X is the affine image of the unit cube under an invertible affine transformation, then the symmetrized function ~f ofany convex function f : X! R satisfies the following double inequality

~f1

nþ 1

Xn

i¼0

v i

!6

ZX

~f ðxÞ dlðxÞ 6 Xj jnþ 1

Xn

i¼0

~f ðv iÞ; ð34Þ

where n ¼ 2d � 1. Equality is attained for all affine functions.Note that in the following Theorem 4.2, the convexity is unnecessary for the function f.

Theorem 4.2. If X is the affine image of the unit cube under an invertible affine transformation and the symmetrized function ~f of fis convex, then the following double inequality holds

f1

nþ 1

Xn

i¼0

v i

!6

ZX

f ðxÞ dx 6Xj j

nþ 1

Xn

i¼0

f ðv iÞ; ð35Þ

where n ¼ 2d � 1. Equality is attained for all affine functions.To conclude this section, we note that the above Theorem may be regarded as an extension of [4, Theorem 2.2], which

states that a convex function satisfies the generalized Hermite–Hadamard inequality (40) on simplices.

5. Applications to Wright-convex functions

The purpose of this section is to prove that the principal results of the preceding section remain true for W-convexfunctions. We first recall some facts and definitions about W-convex functions. A function f : X! R is called Wright-convex(W-convex, for short), if

f ðtxþ ð1� tÞyÞ þ f ðty þ ð1� tÞxÞ 6 f ðxÞ þ f ðyÞ

for every x; y 2 X and t 2 ½0;1�. It is immediately obvious from the definition that any convex function is necessarilyW-convex. The converse is not necessarily true as seen by taking any additive function f, which is not convex. We recall thatan additive function a : Rd ! R is one which satisfies the equality

aðxþ yÞ ¼ aðxÞ þ aðyÞ

for every x; y; xþ y 2 Rd. W-convex functions have a nice characterization in terms of the classical notion of convexity.Indeed, a function f is W-convex if and only if it can be represented in the form f ¼ g þ a, where g is a convex functionon X and a is an additive function from Rd to R, see [12].

We now collect some elementary identities which are trivial consequences of the additivity property.

Lemma 5.1. Let a : Rd ! R be a continuous additive function. Then, for all x in Rd, the function a and its symmetrized function ~asatisfy

að2c � xÞ ¼ 2aðcÞ � aðxÞ; ð36Þ

~aðxÞ ¼ ~aðcÞ ¼ aðcÞ; ð37ÞZX

aðxÞdx ¼ Xj jaðcÞ: ð38Þ

Proof. As can be easily checked, the first equality is obvious. Now, using the additivity of the function a, we immediately getfor any x in Rd:

82 A. Guessab / Applied Mathematics and Computation 243 (2014) 74–82

2aðcÞ ¼ að2cÞ ¼ aðxþ 2c � xÞ ¼ aðxÞ þ að2c � xÞ;

which imply identity (37). The last equality is easily obtained by using identity (37) and making an appropriate change ofvariables. h

Since every W-convex function may be represented as a sum of an additive function and a convex function, we then havethe following as a consequence of Lemma 5.1, Theorems 3.5 and 3.6:

Theorem 5.2. If X is the affine image of the unit cube under an invertible affine transformation, then the symmetrized function ~f ofany W-convex function f : X! R satisfies the following double inequality

~f1

nþ 1

Xn

i¼0

v i

!6

ZX

~f ðxÞ dlðxÞ 6 Xj jnþ 1

Xn

i¼0

~f ðv iÞ; ð39Þ

where n ¼ 2d � 1. Equality is attained for all affine functions.Note that in the following Theorem the convexity is unnecessary for the function f.

Theorem 5.3. If X is the affine image of the unit cube under an invertible affine transformation and the symmetrized function ~f of fis W-convex, then the following double inequality holds

f1

nþ 1

Xn

i¼0

v i

!6

ZX

f ðxÞ dx 6Xj j

nþ 1

Xn

i¼0

f ðv iÞ; ð40Þ

where n ¼ 2d � 1. Equality is attained for all affine functions.

6. Conclusion and a final remark

In this paper, centrally symmetric polytopes are first considered and some of their basic properties have been fullyexplored. A symmetrization technique is proposed, which can be viewed as a multivariate generalization of the approachin the univariate case, recently used by Dragomir [2] and also by El Farissi et al. [3]. This symmetrization technique enablesus to derive new classes of Hermite–Hadamard type inequalities for both the original functions and their symmetrizedfunctions. A remarkable class of centrally symmetric polytopes, such that allows explicit expressions of the integrals ofbarycentric coordinates, is considered in more detail, thus providing explicit formulas for these new inequalities. This paperalso contains some Hermite–Hadamard type inequalities without assuming convexity. We also extend our results to theclass of Wright-convex functions.

Finally, as a concluding remark, we would like to emphasize that another symmetrization procedure was recentlyintroduced in [8]. This technique, which is very different from the one proposed here, is based on permutation enumerationof the symmetric group and ‘generalized’ barycentric coordinates.

Acknowledgment

The author would like to thank the two reviewers for their careful reading of the paper and their helpful comments. Hewould also like to thank financial support from the Volubilis Hubert Curien Program (Grant No. MA/13/286), and Hassan IUniversity, Settat, Morocco for hosting and supporting us during the research stay that led to this collaboration.

References

[1] J.W.S. Cassels, Obituary: Kurt Mahler, Bull. London Math. Soc. 24 (4) (1992) 381–397.[2] S.S. Dragomir, Symmetrized convexity and Hermite–Hadamard type inequalities, RGMIA Research Report Collection 17 (2014) 1-16 (Article 1).[3] A. El Farissi, M. Benbachir, M. Dahmane, An extension of the Hermite–Hadamard inequality for convex symmetrized functions, Real Anal. Exch. 38 (2)

(2012/2013) 469–476.[4] A. Guessab, G. Schmeisser, Convexity results and sharp error estimates in approximate multivariate integration, Math. Comp. 73 (247) (2004) 1365–

1384.[5] A. Guessab, Generalized barycentric coordinates and approximations of convex functions on arbitrary convex polytopes, Comput. Math. Appl. 66 (6)

(2013) 1120–1136.[6] A. Guessab, G. Schmeisser, Sharp integral inequalities of the Hermite–Hadamard type, J. Approx. Theory 115 (2002) 260–288.[7] A. Guessab, Approximations of differentiable convex functions on arbitrary convex polytopes, Appl. Math. Comput. 240 (1) (2014) 326–338.[8] A. Guessab, F. Guessab, Symmetrization, convexity and applications, Appl. Math. Comput. 240 (1) (2014) 149–160.[9] O. Hanner, Intersections of translates of convex bodies, Math. Scand. 4 (1956) 65–87.

[10] K. Kalai, The number of faces of centrally-symmetric polytopes, Graphs Comb. 5 (1) (1989) 389–391.[11] J.A. Kalman, Continuity and convexity of projections and barycentric coordinates in convex polyhedra, Pac. J. Math. 11 (1961) 1017–1022.[12] C.T. Ng, Functions generating Schur-convex sums, in: General inequalities, 5 (Oberwolfach, 1986), Int. Schriftenr. Numer. Math., 80, Birkhuser, Basel,

1987, pp. 433–438.[13] G.M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics, vol. 152, Springer-Verlag, Berlin-Heidelberg-New York-London-Paris-Tokyo-Hong

Kong, 1995. ix + 370 pp. softcover: 3 540 94365 X, 21, hardcover: 3 540 94329 3, 47.