Computers and Mathematics with Applications 66 (2013) 1120–1136

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Computers and Mathematics with Applications

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

Generalized barycentric coordinates and approximations ofconvex functions on arbitrary convex polytopesAllal Guessab ∗

Laboratoire 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

Article history:Received 11 January 2013Received in revised form 19 July 2013Accepted 24 July 2013

Keywords:Barycentric approximation schemesBarycentric coordinatesConvex functionsDelaunay triangulationOptimal approximationPolytopes

a b s t r a c t

In this paper, we study the error in the approximation of a convex function obtainedvia a one-parameter family of approximation schemes, which we refer to as barycentricapproximation schemes. For a given finite set of pairwise distinct points Xn := xini=0 inRd, the barycentric approximation of a convex function f is of the form:

B[f ](x) =

ni=0

λi(x)f (xi),

where λini=0 is a set of barycentric coordinates with respect to the point set Xn. The main

content of this paper is two-fold. The first goal is to derive sharp upper and lower boundson all barycentric coordinates over the convex polytope conv(Xn). The second objectiveof the paper is to exploit the convexity assumption heavily and establish a number ofupper and lower pointwise bounds on the approximation error for approximating arbitraryconvex functions. These bounds depend solely on computable quantities related to thedata values of the function, the largest and smallest barycentric coordinates. For convextwice continuously differentiable functions, we derive an optimal error estimate. We showthat the Delaunay triangulation gives access to efficient algorithms for computing optimalbarycentric approximation. Finally, numerical examples are used to show the success ofthe method.

© 2013 Elsevier Ltd. All rights reserved.

1. Introduction, notations and preliminary results

The basic idea behind barycentric approximation schemes is the following: consider a given finite set of pairwise distinctpoints Xn = xini=0 in P ⊂ Rd, with P = conv(Xn) denoting the convex hull of the point set Xn. We do not make anyassumptions about the distribution of the points in P except that they must not lie entirely in the same straight line. Weare interested in approximating an unknown scalar-valued convex continuous function f : P → R from given functionvalues f (x0), . . . , f (xn) sampled at Xn. In order to obtain a simple and stable global approximation on P , we may consider aweighted average of the function values at data points of the following manner:

B[f ](x) =

ni=0

λi(x)f (xi), (1)

or, equivalently, a convex combination of the data values f (x0), . . . , f (xn). This means that the system of functions λ :=

λini=0 forms a partition of unity, that is:

λi(x) ≥ 0, ∀x ∈ P, i = 0, . . . , n, (2)

∗ Tel.: +33 559674418.E-mail address: allal.guessab@univ-pau.fr.

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A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136 1121

ni=0

λi(x) = 1, ∀x ∈ P. (3)

In addition, we shall also require the set of functions λ to satisfy the first-order consistency conditions:

x =

ni=0

λi(x)xi, (∀x ∈ P). (4)

Wewill call any set of functionsλi : P → R, i = 0, . . . , n, barycentric coordinates if they satisfy collectively, for all x ∈ P , thethree properties (2)–(4). In viewof this propertywe shall refer to the approximation schemes B as barycentric approximation(schemes). The barycentric approximation has many nice properties and can be very useful in some situations. For instance,conditions (3) and (4) together guarantee that affine functions are exactly reproduced by the approximation scheme B. Forsecond-order partial differential equations, approximants that possess constant (3) and linear precision (4) are sufficient forconvergence in a Galerkinmethod [1]. The positivity condition (2) togetherwith the partition of unity property (3), allows usto interpret the barycentric approximation as a weighted average of the data values. This viewpoint is common in computergraphics and geometricmodeling, e.g. in Bézier andB-spline techniques [2,3]. A very useful property of (2) and (3) is that if x isgiven such that 0 ≤ λi(x) ≤ 1 for all i, then x is inside the polytope P . Eqs. (2) and (3) also ensure that approximation operatorB is bounded between the data’s minimum and maximum values: mini f (xi) ≤ B[f ](x) ≤ maxi f (xi). This guarantees astability property of the operator B and it is also a statement of the discrete maximum principle and a requirement for thenumerical discretization of the diffusion equation. Note that B is in general non-interpolatory on the point setXn. However, atthe vertices of the polytope P the function λi associated to the vertex vi has to equal 1 at vi and 0 at all other points inXn\vi,i.e., λi(vj) = δij (δ is the Kronecker delta). This property is the foundation of using barycentric coordinates for interpolationpurposes. Indeed, it immediately implies vertex interpolation for any continuous function: B[f ](vi) = f (vi). In this regard,we should mention that vertex interpolation is satisfied automatically by any positive linear operator, see [4]. Clearly, if f isnonnegative, then so is B[f ]. Hence, the weighted average approximation B preserves the nonnegativity property and thenit belongs to the class of positive operators. We will also see that all functions λi associated to points in the interior of Xnvanish at the boundary of P , such property facilitates the imposition of linear Dirichlet boundary conditions in a Galerkinmethod, [5,6].

It may happen that we know beforehand that the function f to be approximated is convex. To our knowledge, however,subject to the usual convexity, no careful analysis has been designed to be able to provide the correct error estimates forthese barycentric approximation schemes. There is some literature on estimation of functions restricted by convexity. Forinstance, by allowing some smoothness on f to be approximated, say C2(P), it played a crucial role in our research on thedetermination of the ‘best’ (or ‘optimal’) cubature formulas, see [7–12], where the latter problem has been extensivelyreviewed both from the theoretical study as well as the numerical point of view. It is to note that, in the context of onedimensional case, many works have been devoted to the obtention of upper bounds for the discrete Jensen’s inequality, seee.g. [13–15] and references therein.

Therefore, in this paper, we shall exploit the convexity assumption heavily, and establish a number of sharp upper andlower pointwise bounds on the error inherent in barycentric approximation for approximating arbitrary convex functions.

The remainder of this paper, which we have divided into six parts, is organized as follows. In Section 2, we first fixnotation and provide a very brief survey and some relevant refinements on barycentric coordinates on polytopes. We alsogive basic existence results. We then derive in Section 3 sharp upper and lower bounds on all barycentric coordinates overconvex polytopes. In Section 4, we present some of the essential properties of barycentric approximation schemes, and theircorresponding approximation errors. In particular, it is shown that a weak Kronecker-delta property at the boundary holds.This section also provides a characterization of the nonnegativity property of the approximation error on the class of convexfunctions.We also explain in this sectionwhy the framework given in this paperwas limited to the case of polytope domains.In Section 5, we establish a number of upper and lower pointwise bounds on the approximation error for approximatingconvex functions. Our main result is Theorem 5.2, which shows how to control, for any convex function, the pointwise errorat each x ∈ P committed by the approximation B in terms of computable quantities related to the data, the largest andsmallest barycentric coordinates of the point x. In this section, sharp bounds are also derived through the use of the boundson all barycentric coordinates obtained in Section 3. In Section 6, for convex twice continuously differentiable functions, wederive an optimal error estimate.We show that theDelaunay triangulation gives access to efficient algorithms for computingoptimal barycentric approximation. Some numerical results presented in Section 7 illustrate the success of piecewise linearinterpolation using Delaunay triangulation.

2. Generalized barycentric coordinates on polytopes and existence results

In this section we will give the notation we will use, and extend the notion of barycentric coordinates to arbitrarypolytopes. We will also show their existence under very general conditions.

Consider a set of pairwise distinct points Xn = xini=0 in Rd. To simplify the remaining presentation, let us introducesome notations: throughout this paper d and n denote some fixed positive integers. The convex hull, P ⊂ Rd is the smallest(inclusionwise) convex set that contains all the point sets Xn. The convex hull of a finite set of points in some Rd is called

1122 A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136

a (convex) polytope. For simplicity, we drop the qualifier ‘‘convex’’ throughout, since most polytopes under considerationwill be convex. A point x of a polytope P is called an extreme point if there are no two points y = x and z = x in P such thatx is a convex combination of y and z .

For simplices, barycentric coordinates are a very common tool in many computations. Basically, they are defined asfollows: let Xd = v0, . . . , vd be any linearly independent set of d + 1 points in Rd, the simplex T with the set of vertices Xdis the convex hull of Xd, (e.g., a triangle in 2D or a tetrahedron in 3D). Let Ai(x) be the signed volume (or area) of the simplexcreated with the vertex vi replaced by x.

Then the barycentric coordinate functions λi, i = 0, . . . , d of the simplex T with respect to its vertices are uniquelydefined by

λi(x) =Ai(x)vol(T )

, (5)

where vol(T ) will mean the volume measure of T . It is easily seen that each point x of T has a (unique) representation, thatis x =

di=0 λi(x)vi and the barycentric coordinates λ0, . . . , λd are nonnegative affine functions on T . The uniqueness of

this representation allows the weights λi(x) to be interpreted as an alternative set of coordinates for point x, the so-calledbarycentric coordinates.

Note that a simplex is a special polytope given as the convex hull of d + 1 vertices, each pair of which is joined by anedge.

For a convex polytope P ⊂ Rd we will use ‘generalized’ barycentric coordinates. (They are often called generalizedbarycentric coordinates to distinguish them from the original barycentric coordinates, whichwere only definedwith respectto simplices.) While barycentric coordinates are unique for simplices, there are many possible solutions for polygonswith more sides. In recent years, a wide range of generalized barycentric coordinates has been suggested. Two importantexamples of generalized barycentric coordinate functions areWachspress’ coordinates described in [16] and themean valuecoordinates of [2]. Both families produce smooth functions on the polytope P . While being initially described for the 2D case,both have generalizations to 3D. In recent years, the research on barycentric coordinates has been intensified and led to ageneral theory and extensions to higher dimensions [17–21]. Barycentric coordinates are natural coordinates for meshesand have many applications including parameterization [22,23], free form deformations [24,18], finite elements [25]. Foradditional references as well as more details on this topic, refer to the complete web page (http://www.inf.usi.ch/hormann/barycentric/).

The first result on the existence of barycentric coordinates for more general types of polytopes was due to Kalman[26, Theorem 2] (1961). The next lemma is due essentially to Kalman [26]. Our statements are stronger than the onesprovided in [26], but the proof proceeds along the same lines as the proof of Theorem 2 in [26], so we omit it.

Lemma 2.1. Let Xn = x0, . . . , xn be a set of finite pairwise distinct points of Rd and let the polytope P = conv(Xn). Let x∗ bea given point in P with

x∗=

ni=0

λ∗

i xi andn

i=0

λ∗

i = 1, λ∗

i ≥ 0. (6)

Then there exist nonnegative real-valued continuous functions λ0, λ1, . . . , λn defined on P such that

x =

ni=0

λi(x)xi andn

i=0

λi(x) = 1 (7)

for each x ∈ P and λi(x∗) = λ∗

i . Moreover, we can choose the barycentric coordinates λ0, . . . , λn in such a way that one ofthem is convex or concave.

In addition to the assumptions of the previous lemma, we now assume that there are no other points in Xn than the verticesof the polytope P . This situation is of interest when P is a small convex polytope obtained by subdivisions of the originaldomain. The most important fact of the next result is that the barycentric coordinates are linearly independent. In fact, wehave the following.

Lemma 2.2. Let P be a polytope in Rd, v0, v1, . . . , vm its vertices. Then there are linearly independent nonnegative realcontinuous functions on P,

λ = λ0, . . . , λm

defined on P such that

x =

mi=0

λi(x)vi andmi=0

λi(x) = 1 (8)

for each x ∈ P.

A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136 1123

Proof. The existence of a set of continuous barycentric coordinates λ is assured by Lemma 2.1. So it remains to show thatthe set consisting of the functions λ is linearly independent. Linear precision (8) shows in particular that each x may berepresented as a convex combination of v0, v1, . . . , vm. Since each vi is an extreme point of P , we conclude by substitutingx = vi in (8) that λi(vj) = δij. Hence, the functions λ0, . . . , λm satisfy the delta function property. Now, it is easy to see thatthis property implies that the set of functions λ must be linearly independent.

Barycentric coordinates provide a basis for linear finite elements on simplices, and generalized barycentric coordinatesnaturally produce a suitable basis for linear finite elements on general polytopes. Here, the underlying principle is that onetriangulates the polytope into simplices and then use the standard barycentric coordinate functions of these simplices.Interpolation properties of this scheme are well known from the standard analysis of the finite element method oversimplicial meshes, but, as we will see in the next section, this construction will serve as an important point of comparisonwith any set of barycentric coordinates. Indeed, a most common form of approximation in the standard continuous linearfinite elementmethod is piecewise linear (piecewise affine) spline interpolation, see, e.g., Ciarlet [27] amongmany standardreference books on finite element theory. To describe a general setup, let Xn = xini=0 in Rd be a finite subset of P ⊂ Rd withthe property P = conv(Xn). A triangulation T of P with respect to Xn is a decomposition of P into d-dimensional simplicessuch that Xn is the set of all their vertices and the intersection of any two simplices consists of a common lower dimensionalsimplex or is empty. Triangulations of compact convex polytopes exist. Indeed, given any finite set Xn of points that do notall lie on a hyperplane, Chen and Xu [28, p. 301] describe a lifting-and-projection procedure which results in a triangulationof the convex hull of Xn. Let S1, . . . Sl be the simplices of T and let Ni be the set of all integers j such that xi is a vertex of Sj.If x ∈ Sj and j ∈ Ni, then we denote by λij(x) the barycentric coordinate of xwith respect to xi for the simplex Sj. It is easilyverified that if x ∈ Sj ∩ Sk, then λij(x) = λik(x) if j, k ∈ Ni and λij(x) = 0 if j ∈ Ni, k ∈ Ni. Therefore, setting

lfei (x) :=

λij(x) if x ∈ Sj and j ∈ Ni0 otherwise

for i = 0, . . . , n, we trivially obtain a set of well-defined barycentric coordinates, which are a generalization of barycentriccoordinates when P is a simplex. We list some basic properties of the functions

lfe0 , . . . , lfen (9)

of which the following are particularly relevant to us.

(1) They are well-defined, piecewise linear and nonnegative real-valued continuous functions.(2) The function lfei has to equal 1 at xi and 0 at all other points in Xn \ xi, i.e., l

fei (vj) = δij (δ is the Kronecker delta).

3. Sharp upper and lower bounds on all barycentric coordinates

Floater, Hormann and Kós [29] derived sharp upper and lower bounds on all barycentric coordinates on convex polygonsin the plane. The main objective of this section is to show that the results of [29] can be extended in a more general way forarbitrary polytopes in higher dimensions.

The important property that will be used later is that the polytope

P = conv (Xn = x0, . . . , xn)

can also be characterized by means of a finite number of linear halfspaces.

P =x ∈ Rd

: hk(x) = ⟨ak, x⟩ + bk ≥ 0, k = 0, . . . ,m, (10)

here ak ∈ Rd and bk ∈ R, see, e.g., [30]. Let (nv+ 1) be the number of vertices of P . We will use the notation V (P) :=

v0, v1, . . . , vnv to denote the set of all vertices of P .Throughout this section P1 is the class of all polynomials in d real variables of degree at most 1, also called the class of

affine functions on Rd, and λ := λ0, λ1, . . . , λn an arbitrary but fixed set of barycentric coordinates with respect to Xn.The next lemma provides an upper bound for each barycentric coordinate associated to a point of Xn, which is in the

interior of P . Here, hj, j = 0, . . . ,m, are the hyperplanes defining the polytope P , see (10).

Lemma 3.1. Let λ0, λ1, . . . , λn be any barycentric coordinate system with respect to Xn = x0, . . . , xn. Then, for anyxi ∈ Xn ∩ int(P) the associated barycentric coordinate λi satisfies the bounds:

0 ≤ λi(x) ≤ min0≤j≤m

hj(x)hj(xi)

≤ 1, (∀x ∈ P). (11)

Proof. We first establish that any set of barycentric coordinates by definition reproduce affine functions. Indeed, let l ∈ P1,then there exist a vector a ∈ Rd and a real number b such that l can be expressed as follows:

l(x) = ⟨a, x⟩ + b,

1124 A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136

Fig. 1. Partition of the polytope P into two polytopes.

where ⟨., .⟩ denotes the usual inner product in Rd. Then, assuming (3) and (4), it follows thatn

i=0

l(xi)λi(x) =

ni=0

(⟨a, xi⟩ + b) λi(x)

=

a,

ni=0

λi(x)xi

+ b

ni=0

λi(x)

= ⟨a, x⟩ + b

= l(x), (∀x ∈ P). (12)

Therefore, for all l ∈ P1,

B[l] = l. (13)

This shows that B reproduces exactly all affine functions. Now take any point xi in Xn ∩ int(P), and let λi its associatedbarycentric coordinate. We next show that λi satisfies the following bounds:

λi ≤hj

hj(xi), (j = 0, . . . ,m).

Indeed, making use of the reproduction property of affine functions and the nonnegativity of hjhj(xi)

, we conclude that, for anyj = 0, . . . ,m,

λi(x) = λi(x)hj(xi)hj(xi)

≤

nk=0

λk(x)hj(xk)hj(xi)

= B

hj

hj(xi)

(x)

=hj(x)hj(xi)

. (14)

Hence, any barycentric coordinate associated to a point in Xn ∩ int(P) must satisfy the required inequality (11), asclaimed.

For the barycentric coordinates associated with each vertex, we get more precise bounds. Given a vertex vi of the polytopeP . First of all, let us denote by P1

i and P2i the two polytopes defined by the convex hull of Xn \vi and P \ int(P1

i ) respectively.Then, it is easily seen that P1

i and P2i are two smaller polytopes of P , and they form a partition of P . As was shown in the

example of Fig. 1, the set P2i , in general, need not be convex. Let us denote by lij, j = 0 . . . ,mi the hyperplanes defining the

common facets of P1i and P2

i . These hyperplanes with the additional normalization condition lij(vi) = 1 play a predominantrole in determining lower bounds on any barycentric coordinates.

Let us make now some comments on the hyperplanes lij.

A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136 1125

Remark 3.2. The hyperplanes lij enjoy many truly remarkable properties.

(i) Since lij, j = 0, . . . ,mi, are supporting hyperplanes of P1i , then the polytope P1

i is entirely contained in one of the two closedhalfspaces determined by lij. Hence, it follows, in particular, from the fact that lij(vi) = 1,

lij(xk) ≤ 0, k = 0, . . . , n, k = i. (15)

(ii) For each i = 0, . . . , nv , the function associated with vertex vi defined by

li := maxj=0,...,mi

0, lij

(16)

is obviously a nonnegative convex function on P, that it takes the value 1 at the vertex vi. We note that it also follows from(15) that

li(xk) = 0, k = 0, . . . , n, k = i. (17)

Hence li satisfies the delta-property, and then, in particular, li vanishes at all vertices of P1i . Thus, it follows from nonnegativity

and convexity of li, that it vanishes entirely on the polytope conv (Xn \ vi) := P1i .

After these preparations, we can prove the following result.

Lemma 3.3. Under the above notations, any barycentric coordinate λi associated with a vertex vi satisfies for all x ∈ P thebounds:

0 ≤ li(x) = maxj=0,...,mi

0, lij(x)

≤ λi(x) ≤ Li(x) = min

j∈∆i

hj(x)hj(vi)

≤ 1, (18)

where ∆i is the set of all indices j such that the hyperplane hj does not vanish at the vertex vi.

Proof. Fix a vertex vi of the polytope P , and let λi its associated barycentric coordinate. The upper bound in (18) may beproved in a similar way by following the proof of Lemma 3.1, so we omit it. To prove the lower bound, we first note, asmentioned in Remark 3.2, (ii), that the function li vanishes on P1

i , then we may assume that x ∈ P2i , since otherwise the

lower bound is automatically satisfied. Hence, let us pick x any point in P2i . Then, since lij is an affine function

lij(x) =

nk=0

λk(x)lij(xk)

= λi(x)lij(vi) +

nk=0,k=i

λk(x)lij(xk). (19)

Hence, in view of (15), lij(x) can be bounded by

lij(x) ≤ λi(x). (20)

This implies the required lower bound and completes the proof of the lemma.

Lemma 3.3 extends [29, Proposition 1] to any dimension and as immediate consequences of the last two lemmas, we havethe following general results.

Corollary 3.4. Any set of barycentric coordinates satisfies the following.

(i) The barycentric coordinates associated to each point belonging to the interior of the polytope P vanish on facets of P.(ii) The barycentric coordinates associated to the vertices satisfy the delta property.

Some of these properties in different forms are known in many contexts. For instance, as a consequence of Corollary 3.4Bézier curves pass through the end control points and are tangent to the end control-polygon edges, see [2,31].

Fix vi a vertex of P , and for each j ∈ 0, . . . ,mi, let Qji be the polytope defined by conv

F ji ∪ vi

, where F j

i is the facet

defining hyperplane lji. With this notation it is clear that these polytopes contain vi as a vertex and form a partition of P2i . The

polytope Q ji can be triangulated using no new vertices, and such that any simplex in T has vi as a vertex, see [32], see Fig. 2.

Note that the uniqueness of the barycentric coordinates for simplices implies that the barycentric coordinates associated tothe vertex vi with respect to any simplex of T is exactly the equation of the hyperplane lij.

The next result shows that no better bounds than li and Li can be obtained for any set of barycentric coordinates. In viewof this fact, we call them sharp lower and upper bounds. Indeed, wewill showmore precisely that there is a set of barycentriccoordinates for which these bounds are themselves i-th barycentric coordinates. We say that a function λi : P → R is an

1126 A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136

Fig. 2. An example of a two-dimensional triangulation of the polytope P2i into simplices.

i-th barycentric coordinate if it is the i-th member of some set of barycentric coordinates. With the help of Lemma 2.1, wecan prove the following.

Proposition 3.5. The functions li and Li associated to any vertex vi are i-th barycentric coordinates.

Proof. Given a vertex vi of P. The main ingredient in the proof is the key Lemma 2.1, which says that there exists a set ofbarycentric coordinates β0, β1, . . . , βn such that the barycentric coordinate βi associated to vi is a convex function (or aconcave function). We will now show that under the convexity assumption of βi, the two functions βi and li are identicallyequal on P . Corollary 3.4, (ii) implies that βi must satisfy the delta property, that is

βi(vj) = δij, (21)

and consequently βi vanishes on the polytope conv (Xn \ vi) := P1i . Hence, βi and li are equal on P1

i . Let now x be any pointof P2

i = P \ int(P1i ), then it belongs to at least one simplex S ij of the triangulation T of the polytope Q j

i . Since βi satisfies thedelta property at the vertices of any simplex of S ij , it follows immediately from Jensen’s inequality that βi(x) ≤ lij(x), here,we have taken into account the fact that the barycentric coordinate with respect to any simplex of T associated to the vertexvi is the hyperplane lij. Now, this immediately implies that βi(x) ≤ li(x). Finally, this together with the opposite inequalityproved in Lemma 3.3, satisfied by any set of barycentric coordinates, shows that βi = li on P2

i .We can show in a completely analogous way that if βi is concave then Li = βi. Thus we have proved Proposition 3.5.

The result of Proposition 3.5 is precisely the result of Floater, Hormann and Kós [29, Proposition 2.4] derived on convexpolygons in the plane.

The next result shows that the reverse implication of Lemma 3.3 holds. Indeed, applying the same argument as in thecase of polygons in [29], we can show that all functions between the upper and lower bounds, obtained in Lemma 3.3, arei-th barycentric coordinates.

Proposition 3.6. A function λi : P → R is an i-th barycentric coordinate associated to a vertex if and only if the following boundshold for all x in P.

li(x) ≤ λi(x) ≤ Li(x). (22)

Proof. Fix i ∈ 0, . . . , n, and take the vertex vi of the polytope P .Wehave already shown, in Lemma3.3, that any barycentriccoordinate λi associated to vi must satisfy the bound given in (22). Thus, the bounds (22) are necessary for λi to be an i-thbarycentric coordinate with respect to a vertex. To show that the bounds are also sufficient, note first that they immediatelyimply

λi(x) = (1 − αi(x))li(x) + αi(x)Li(x), (∀x ∈ P), (23)

for some function αi : P → [0, 1]. By Proposition 3.5 there exist two sets of barycentric coordinates λ0, λ1, . . . , λn andβ0, β1, . . . , βn such that

λi(x) = li(x)βi(x) = Li(x).

Now define the set of functions γ0, γ1, . . . , γn by

γk(x) = (1 − αi(x))λk(x) + αi(x)βk(x), (∀x ∈ P). (24)

A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136 1127

It is easy to check that γ0, γ1, . . . , γn form a set of barycentric coordinates and since λi = γi, this means that λi is indeedan i-th barycentric coordinate.

Additional comparisons of barycentric functions can be found in the survey papers of Cueto et al. [5], Sukumar andTabarraei [33].

4. General properties of barycentric approximation schemes

Barycentric approximation schemes and their corresponding approximation errors share a number of desirableproperties. The next section enumerates these properties. Let us denote the set of all continuous functions from the polytopeP to R by C(P). The subset of C(P) formed by all continuous convex functions on P is denoted by K(P). We will also denoteby Rn the subspace of C(P) spanned by the set of barycentric coordinates λ = λ0, λ1, . . . , λn. As a consequence of (3) and(4), the linear space Rn contains P1 as a subspace.

We now state without proof some essential properties of barycentric approximation schemes.

Theorem 4.1. The barycentric approximation scheme B has the following properties.(i) It maps C(P) into Rn.(ii) It is nonexpansive with respect to the uniform norm ∥.∥∞, in the sense that if f , g ∈ C(P) then

∥B[f ] − B[g]∥∞ ≤ ∥f − g∥∞ . (25)(iii) It cannot have a second-order approximation.

The following statements, which easily follow from a simple calculation, give us some important properties relevant to theincurred approximation error, between f and its barycentric approximation. In the following wewill often use the notation:

E[f ](x) = E[f , λ,Xn] :=

ni=0

λi(x)f (xi) − f (x). (26)

Theorem 4.2. The error approximation E has the following properties.(i) It vanishes over affine functions, this means that B is a first-order approximation scheme, i.e., it reproduces exactly all affine

functions.(ii) It is nonnegative in K(P). This means that B approximates convex functions from above.(iii) For any real continuous function f on P, the error E[f ] vanishes at the vertices. Hence, B satisfies the vertex interpolation:

B[f ](vi) = f (vi), for all vertices of P.

LetΩ be a compact convex subset ofRd and let p := pini=0 be a partition of unity, that is a collection of continuous functionsfrom Ω to the unit interval [0, 1] and whose values sum the unity for all x ∈ Ω . Let x0, . . . , xn be a set of distinct pointsin Ω . In a partition of unity method, an approximation of a function f defined on Ω is given by

Apu[f ](x) =

ni=0

pi(x)f (xi). (27)

In the case that p forms a set of barycentric coordinates, Theorem 4.2 shows that the nonnegativity property (ii) is satisfiedby the approximation error Apu

[f ] − f on K(P).The next result shows that, in a certain sense, the reverse implication also holds. The main point of what follows is that

the nonnegativity property on K(P) of the approximation error E can be characterized in terms of linear precision of thebarycentric coordinates. Here, and in what follows, A(P) will denote the space of all affine real functions on P .

Theorem 4.3. Let Ω be a compact convex subset of Rd with positivemeasure. Assume that there are n+1 points x0, . . . , xn ∈ Ω ,and a partition of unity p0, . . . , pn on Ω . Then, the following statements are equivalent.(i) p0, . . . , pn is a barycentric coordinate system.(ii) The error E is nonnegative on K(P).(iii) The error E vanishes on A(Ω).

Proof. We have already shown the implication (i) implies (ii) in Theorem 4.2, (ii). For the implication (ii) implies (iii), weuse the fact that affine functions and their opposites belong to K(Ω). Then (ii) implies that for any affine function lwe haveE[l] ≥ 0 and E[−l] ≥ 0. Hence, by virtue of linearity of the operator E we deduce the vanishing property of E on A(P). Finally,to prove that (iii) implies (i) it suffices to consider the usual projection functions e1, . . . , ed, ei : x = (x1, . . . , xd) → xi.

5. Pointwise error estimations

In this section, we will discuss the problem of finding upper and lower bounds for the approximation error betweenany continuous convex function and its barycentric approximation scheme defined by (1). Here, the error induced by theapproximation process will be estimated by using a pointwise error measure. For a given convex function f , effective upper

1128 A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136

and lower bounds for E[f ] at a point x can be given in terms of the data values of f , the largest and smallest barycentriccoordinates of x. These bounds are presented in Eqs. (34) and (50) of Theorems 5.2 and 5.4, respectively.

More precisely, assume we are given a set λ = λ0, λ1, . . . , λn of barycentric coordinates with respect to a fixed set of(n + 1) pairwise distinct points Xn = xini=0 of Rd, we are interested in estimating, for each x ∈ P , the quantity:

E[f ](x) =

ni=0

λi(x)f (xi) − f (x), (∀ f ∈ K(P)). (28)

Before stating any error estimates, we fix some notation. Throughout, let V (P) := v0, v1, . . . , vnv denote the set of verticesof P . If there is no confusion, we will continue to simply denote E[f , λ,Xn] by E[f ].

The centroid w[Xn] of the polytope P = conv(Xn) with respect to Xn is defined as the average of the points defining thepolytope. Therefore, we have:

w[Xn] =1

n + 1

ni=0

xi, (29)

which we often write as w when Xn is clear from context. We note that w is always located inside P . In what follows, wewill let e[f ](w) denote

e[f ](w) :=1

n + 1

ni=0

f (xi) − f (w). (30)

This fundamental quantity will play a significant role in estimating the error E. The reader may verify that under theconditions

λi(w) =1

n + 1, i = 0, . . . , n, (31)

e[f ](w) coincides with E[f ](w). Recall that Lemma 2.1 says that the set of barycentric coordinates λ can be chosen such thatthe equalities given in Eq. (31) hold.

If f ∈ K(P), then Theorem 4.2, (ii), asserts that 0 ≤ E[f ]. The lower bound zero is of global nature since it holds forany x ∈ P, f ∈ K(P) and any barycentric coordinate system. We observe, under the assumption that f is convex, that thefollowing upper bound holds trivially.

E[f ](x) ≤ maxy∈P

f (y) − f (x), (∀ f ∈ K(P)). (32)

We have to work a little more in order to improve on the bound given by (32). Indeed, in the subsequent analysis, we shallfirst prove a remarkable fact that the error E[f ] made in approximating f ∈ K(P) at any point x ∈ P can be controlled bythe error e[f ] at the centroid of Xn. Indeed, the next result gives an upper global bound (depending solely on e[f ] at w) andit does not depend on the set of barycentric coordinates used by the approximation operator B.

Theorem 5.1. With the above notation, if f is a continuous convex function on P, then we have:

0 ≤ E[f ](x) ≤ (n + 1)e[f ](w), (∀x ∈ P). (33)

Proof. To show this we first note that the centroidw can be written as

w =1

n + 1

ni=0

xi

=1

n + 1

n

i=0

xi − x + x

=1

n + 1

n

i=0

xi −n

i=0

λi(x)xi + x

=

ni=0

1 − λi(x)n + 1

xi +1

n + 1x,

which is clearly a convex combination of points of P . Now, Jensen’s inequality immediately implies

(n + 1)f (w) ≤

ni=0

(1 − λi(x)) f (xi) + f (x)

=

ni=0

f (xi) − E[f ](x),

which is clearly equivalent to the desired inequality.

A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136 1129

As the reader may verify, the bound given in Theorem 5.1 is better than that given in (32). The next result gives more preciselower and upper bounds for the error E. To do so, it will be convenient to use two important functions. For any point x ∈ Pwith barycentric coordinates λ0(x), . . . , λn(x), we define the largest and smallest barycentric coordinates as follows:

λmax(x) := maxi=0,...,n

λi(x)

λmin(x) := mini=0,...,n

λi(x).

For any given x ∈ P and f ∈ K(P), Theorem 5.2 shows how to control the error E[f ](x) in terms of the largest and smallestbarycentric coordinates of x and the error e[f ](w).

Theorem 5.2. If f is a continuous convex function on P, then, for all x in P, we have:

(n + 1)λmin(x)e[f ](w) ≤ E[f ](x) ≤ (n + 1)λmax(x)e[f ](w). (34)

Proof. Pick any x in P . We distinguish three cases.

• Case I: λmax(x) ≤1

n+1 . Then, λi(x) ≤1

n+1 , i = 0, . . . , n, therefore λi(x) =1

n+1 , i = 0, . . . , n, since we haveni=0

1n+1 − λi(x)

= 0. This means that x = w, so inequality (34) obviously holds.

• Case II: 1n+1 ≤ λmin(x). Then, the same argument shows that λi(x) =

1n+1 , i = 0, . . . , n, x = w, and again inequality

(34) obviously holds.• Case III: λmmin(x) < 1

n+1 < λmax(x).

To simplify notation, let us denote

l = (n + 1)λmin(x).

Since 0 ≤ l < 1, we can write x as a convex combination of x0, . . . , xn, namely,

x = lw + (1 − l)y, (35)

where

y =

ni=0

λi(x) − λmin(x)1 − l

xi. (36)

It is not hard to check also that y is a convex combination of the points x0, . . . , xn. Hence, by using two times Jensen’sinequality we get

f (x) ≤ lf (w) +

ni=0

(λi(x) − λmin(x)) f (xi)

= lf (w) +

ni=0

λi(x)f (xi) −l

n + 1

ni=0

f (xi),

which is equivalent to the left inequality in (34).To show the right-hand side inequality in (34), once again, we can express the centroidw in the form:

w =1rx +

1 −

1r

z, (37)

where, for ease of notation, we set

z =

ni=0

λmax(x) − λi(x)r − 1

xi, (38)

and

r = (n + 1)λmax(x).

Note that, since 1n+1 < λmax(x) then 0 < 1

r < 1 and therefore z and w are written as convex combinations of points of P .Hence, by Jensen’s inequality applied to (37), we get after straightforward calculations.

f (z) ≥r

r − 1f (w) −

1r − 1

f (x). (39)

1130 A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136

On the other hand, we have

rn + 1

ni=0

f (xi) =

ni=0

λmax(x)f (xi)

=

ni=0

λi(x)f (xi) + (r − 1)n

i=0

λmax(x) − λi(x)r − 1

f (xi)

≥

ni=0

λi(x)f (xi) + (r − 1)f

n

i=0

λmax(x) − λi(x)r − 1

xi

=

ni=0

λi(x)f (xi) + (r − 1)f (z).

Now, making use of (39), we rewrite the above inequality as follows:

rn + 1

ni=0

f (xi) ≥

ni=0

λi(x)f (xi)f (xi) + rf (w) − f (x). (40)

Finally, short calculations based on the definition of r establish the right hand side of the inequality in (34). This validatesthe two bounds given in (34).

There are two points that distinguish this result from what has been reported before. First one observes, the simple formof the upper bound that appears at the right hand side of the inequality (34) is better than those given by (33). Second,Theorem 5.2 yields a significantly better lower bound than the bound 0 provided by Theorem 5.1.

It is generally not necessary to use all the data points to estimate the error E. Instead, we can use the data values at thevertices. Let us describe this technique by assuming, for instance, that we only know the values at the vertices of the convexfunction to be approximated. In this setting, we introduce a particularly useful notation which will be used throughout:

ev[f ](wv) :=

1nv + 1

nvi=0

f (vi) − f (wv), (41)

where nv+ 1 denotes the number of vertices of the polytope P . As before, we shall also use the notation

wv=

1nv + 1

nvi=0

vi, (42)

to denote the centroid of the polytope P with respect to its vertices.From now on the symbol Ev will be reserved exclusively to denote the approximation error when the set of points Xn is

constituted only by vertices of the polytope P . Thus, we have

Ev[f ](x) :=

nvi=0

λvi (x)f (vi) − f (x)

hereλv0, λ

v1, . . . , λ

vnv

is a given set of barycentric coordinates of x with respect to the set of vertices v0, . . . , vnv . With

these conventions in mind, the next lemma tells us how the approximation error E[f ], at any x ∈ P , can be controlled byEv

[f ] := Ev[f , αv, V (P)], where αv is a set of barycentric coordinates with respect to V (P). Indeed we have the following

result.

Lemma 5.3. Fix λ := λ0, λ1, . . . , λn a set of barycentric coordinates with respect to Xn = x0, . . . , xn. Then, there exists a setof barycentric coordinates αv

:=αv0 , α

v1 , . . . , α

vnv

with respect to v0, . . . , vnv , such that, for all continuous convex functions

f , and x in P, we have:

E[f , λ,Xn](x) ≤ Ev[f , α, V (P)](x). (43)

Proof. Pick any set of barycentric coordinatesλv0, λ

v1, . . . , λ

vnv

with respect to V (P). Clearly, the points x0, . . . , xn are a

convex combination of the vertices of P , that is,

xi =

nvj=0

λvj (xi)vj, i = 0, . . . , n.

A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136 1131

Hence, by the convexity of f ,

ni=0

λi(x)f (xi) =

ni=0

λi(x)f

nvj=0

λvj (xi)vj

≤

ni=0

λi(x)

nvj=0

λvj (xi)f (vj)

=

nvj=0

αvj (x)f (vj), (44)

where

αvj (x) :=

ni=0

λi(x)λvj (xi), j = 0, . . . , nv.

It is easy to check that

αv0 , α

v1 , . . . , α

vnv (45)

form a set of barycentric coordinates with respect to v0, . . . , vnv . Finally, we get the desired result by subtracting f (x)from both sides of inequality (44).

Combining the previous lemma and Theorem 5.2, we have the following.

Theorem 5.4. With the above notation, there exists a set of barycentric coordinates αv:=

αv0 , α

v1 , . . . , α

vnv

with respect to

v0, . . . , vnv , such that, for all continuous convex functions f , and x in P, we have:

(nv+ 1)λmin(x)ev

[f ](wv) ≤ E[f ](x) ≤ (nv+ 1)αmax(x)ev

[f ](wv), (46)

where

αmax(x) := maxj=0,...,nv

αvj (x). (47)

Proof. Lemma 5.3 says that

E[f ](x) ≤ Ev[f ](x). (48)

Then, we readily see that the right hand side inequality now follows directly from Theorem 5.2, by taking the set ofbarycentric coordinates given in (45).

To establish the left-hand side inequality of (50), it follows from Theorem 5.2 that it is enough to show that

(nv+ 1)

(n + 1)ev

[f ](wv) ≤ e[f ](w). (49)

To see this, we write the centroidw as follows:

w :=1

n + 1

ni=0

xi

=

n−nvi=1

yin + 1

+nv

+ 1n + 1

wv,

where yi, i = 1, . . . , n− nv are elements of Xn located in int(Xn). Obviously, this representation is a convex combination ofelements of P. Therefore, as a consequence of the convexity of f ,

f (w) ≤1

n + 1

n−nvi=1

f (yi) +nv

+ 1n + 1

f (wv),

=1

n + 1

n

i=0

f (xi) −

nvi=0

f (vi)

+

nv+ 1

n + 1f (wv).

Finally, a simple transformation shows that the last inequality is exactly (49) below.

1132 A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136

We now use sharp upper and lower bounds, on all barycentric coordinates obtained in Lemma 3.3, to derive new bounds onthe approximation error Ev . To this end, let li and Li be defined as in Lemma 3.3. For each x in P define

Lmax(x) := maxi=0,...,nv

Li(x)

lmin(x) := mini=0,...,nv

li(x).

As an immediate consequence of Theorem 5.4 and Lemma 3.3, we obtain the following.

Theorem 5.5. If f is a continuous convex function on P, then, for all x in P, we have:

(nv+ 1)lmin(x)ev

[f ](wv) ≤ Ev[f ](x) ≤ (nv

+ 1)Lmax(x)ev[f ](wv). (50)

6. The case of a convex function f ∈ C2(P)

Clearly the lower and upper bounds obtained in Theorems 5.1 and 5.2 depend on a choice of the set of barycentriccoordinates λ := λ0, λ1, . . . , λn used to define the barycentric approximation B. This raises the question of what wouldbe a best choice of λ. In this section, we will restrict our study to convex functions f ∈ C2(P).

For twice differentiable functions f : P → R, we denote by

H[f ](x) :=

∂ f

∂xi∂xj

i,j=1,...,d

the Hessian matrix of f at x and introduceD2f := sup

x∈Psup

y∈Rd,∥y∥=1

yTH[f ](x)y . (51)

All vectors are understood as column vectors, thus xT , or the transpose of x, is a row vector.The following result is a simple consequence of the classical Taylor’s formula for multivariate real valued functions.

Proposition 6.1. Suppose f is a C2-convex function on P. Then, for all x in P, we have

0 ≤ E[f , λ,Xn](x) ≤

D2f

2E[∥.∥2 , λ,Xn](x)

=

D2f

2

ni=0

λi(x) ∥x − xi∥2 . (52)

The equality is attained for all functions of the form f (x) = a(x) + c, where c ∈ R and a is any affine function.

In view of the error estimate (52), it seems desirable to select the set of barycentric coordinates λ for whichE[∥.∥2 , λ,Xn] :=

ni=0 λi ∥x − xi∥2 is small. Hence, wewant to solve the following constrained optimization problem (Px):

for a fixed but arbitrary x ∈ P

minimizeλ

E[∥.∥2 , λ,Xn](x),

subject to the constraints λi ≥ 0, i = 0, . . . , n,n

i=0

λi = 1,

x =

ni=0

λixi.

Since the parametric optimization problem (Px) depends on x, its solution is also a function of x. We shall call a set ofbarycentric coordinatesλ (global) optimal if for each x ∈ P ,λ0(x), λ1(x), . . . , λn(x) is a solution of (Px). Given a triangulationT of the point set Xn, then the point x belongs to at least one simplex Sx of the triangulation T . The set of constraints of (Px)provides (d + 1) constraints, hence, n − d weights can be chosen freely. One possible choice of weights would be to takethe barycentric coordinates of the point x with respect to the simplex Sx and set the remaining (n − d) weights to zero. Ofcourse, the resulting choice is not necessarily optimal. We will show that there exists a triangulation for which this choiceis optimal. A very natural triangulation DT (P) of P , that uses only the points of Xn as triangulation vertices and such thatno point in Xn is inside the hypercircumsphere of any simplex of the triangulation. Such a triangulation exists and it iscalled a Delaunay triangulation of P with respect to Xn and it can be obtained as the geometric dual of the Voronoi diagramof Xn, see, e.g., [34]. We recall from Section 2 that every Delaunay triangulation generates a set of (linear finite element)barycentric coordinates. The following theorem is an immediate consequence of a known extremal property of Delaunaytriangulation.

A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136 1133

Fig. 3. Shows two different Delaunay triangulations of a two-dimensional square.

Theorem 6.2. Every set of barycentric coordinates, which is generated by a Delaunay triangulation, is optimal.

Proof. Let the function g be defined by

g(x) = minλ

E[∥.∥2 , λ,Xn](x), (∀x ∈ P).

By [35, Lemma 10], formulated in our notation and for our situation, at g(x), for a fixed point x, the only nonzero values ofλi occur for the vertices of the Delaunay simplex containing the point x. Thus, a necessary condition for minimizing the lastexpression is that λ is a set of barycentric coordinates, which is generated by a Delaunay triangulation. This completes theproof.

Note that if P is a simplex in Rd, then n = d and the problem (Px) yields for each x a unique solution for the λi. Forn > d + 1, which is the case of interest in this paper, the problem (Px) forms an under-determined system. If pointsx0, . . . , xn are in general position, no d + 1 points of Xn lie on a common hyperplane in Rd and no d + 2 points lie on acommon hypersphere, then Delaunay triangulation is unique. An inspection of the proof of Theorem 6.2 shows that in thiscase, the global unique solution is described by the set of (linear finite element) barycentric coordinates generated by theunique Delaunay triangulation.

When the points are not in general position, the optimization problem above does not have a unique optimal solution.Note that in this case, the problem has an infinity of solutions, indeed, a convex combination of global solutions of (Px) isalso a solution. So the optimal solutions to (Px) form a convex set. It will be interesting to have a way of comparing optimalsolutions and selecting favorable ones. This will be discussed in a forthcoming paper.

It should be noted that not every set of optimal barycentric coordinates is generated by a Delaunay triangulation. In fact,consider a two-dimensional square S, Fig. 3 shows twodifferentDelaunay triangulations are possible for S. Nowevery convexcombination of these solutions provides a set of optimal barycentric coordinateswhich are not generated by a triangulation.

We should mention that for numerical aspects there exist efficient algorithms for computing a Delaunay triangulationwith respect to a finite point set Xn whose convex hull is P , see [34]. Fig. 4(a) shows 21 equally spaced nodes on each edgeof the boundary of square D := [0, 1] × [0, 1] and 361 scattered points randomly selected in D. Fig. 4(b) shows Delaunaytriangulation based on the data points given in Fig. 4(a). Once we have such a triangulation, we can obtain the set of optimalbarycentric coordinates. Finally, for the corresponding L1 problem we refer to [8].

7. Numerical experiments

Given a set of scattered data xi, yi, fiNi=1, which are assumed to be sampled from a convex function f : Ω ⊂ R2→ R.

Taking the N scattered points as nodes, an optimal triangulation mesh, T , is constructed in domain Ω using the Delaunaytriangulation method. In this section, we present some numerical tests to illustrate the practical performance of thecorresponding linear interpolating scheme developed in Section 6.With this aim, we approximate two test convex functionsusing randomly scattered data points in the indicated domain. Although our final findings and conclusions hold for manymore examples we only present here two typical examples.

Example 7.1. We take the following convex function from [36]:

f (x, y) := x4 + y4.

The data points are generated from the above function with the restriction of domain D := [0, 1] × [0, 1]. Fig. 5(a) isDelaunay triangulation based on 21 equally spaced nodes on each edge of the boundary of square D and 216 nodes in thesquare. The nodes in the domain are placed randomly selected from Dwhile the nodes on the boundary are equally spaced.Fig. 5(b) presents the graph of f . Fig. 5(c) describes the graph for the linear interpolation of scattered data generated fromthe function f .

1134 A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136

Fig. 4. Delaunay triangulation for a 2-D set of points.

Fig. 5. (a) Delaunay triangulation for the domain of f , (b) the graph of f and (c) the graph for the linear interpolation of the data generated from f .

Example 7.2. In the second example the data points are generated from the following test function taken from [37]:

g(x, y) = x3 + 5(y2 − 0.6)2 + 1.

As it is mentioned in [37], the data points in this example belong to a surface that models part of a car. The data points aregenerated from the above function with the restriction of domain D := [0, 1]× [0, 1], with data values taken from functiong . Fig. 6(a) shows the Delaunay triangulation based on 21 equally spaced nodes on each edge of the boundary of square Dand 216 nodes randomly selected in the square. Fig. 6(b) is the graph of g while Fig. 6(c) shows the linear interpolation ofscattered data generated from the function g.

A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136 1135

Fig. 6. (a) Delaunay triangulation for the domain of g , (b) the graph of g and (c) the graph for the linear interpolation of the data generated from g .

From Figs. 5 and 6 it is clear that the convexity of f and g has been preserved and there are no visual differences betweenthe test functions and their linear interpolants.

Acknowledgments

The author would like to thank the two referees for their useful proposals. This paper is dedicated to Professor Jean-LouisGout on the occasion of his retirement.

References

[1] G. Strang, G. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.[2] M.S. Floater, Mean value coordinates, Computer Aided Geometric Design 20 (2003) 19–27.[3] A. Schrijver, Theory of Linear and Integer Programming, Wiley, 1987.[4] A. Guessab, O. Nouisser, G. Schmeisser, Enhancement of the algebraic precision of a linear operator and consequences under positivity, Positivity 13

(4) (2009) 693–707.[5] E. Cueto, N. Sukumar, B. Calvo, M.A. Martnez, J. Cegoino, M. Doblar, Overview and recent advances in natural neighbor Galerkin methods, Archives of

Computational Methods in Engineering 10 (4) (2003) 307–384.[6] J.M. Peña, B-splines and optimal stability, Mathematics of Computation 66 (1997) 1555–1560.[7] A. Guessab, O. Nouisser, J. Pečarić, A multivariate extension of an inequality of Brenner–Alzer, Archiv der Mathematik 98 (3) (2012) 277–287.[8] A. Guessab, G. Schmeisser, Negative definite cubature formulae, extremality and Delaunay triangulation, Constructive Approximation 31 (2010)

95–113.[9] A. Guessab, G. Schmeisser, Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations, Advances in

Computational Mathematics 32 (2010) 25–41.[10] A. Guessab, O. Nouisser, G. Schmeisser, A definiteness theory for cubature formulae of order two, Constructive Approximation 24 (2006) 263–288.[11] A. Guessab, G. Schmeisser, Convexity results and sharp error estimates in approximate multivariate integration, Mathematics of Computation 73

(2004) 1365–1384.[12] A. Guessab, G. Schmeisser, Sharp error estimates for interpolatory approximation on convex polytopes, SIAM Journal on Numerical Analysis 43 (3)

(2005) 909–923.[13] S. Dragomir, A converse of the Jensen inequality for convex mappings of several variables and applications, Acta Mathematica Vietnamica 29 (1)

(2004) 77–88.[14] M. Klaricic Bakula, J. Pecaric, J. Peric, On the converse Jensen inequality, Applied Mathematics and Computation 218 (2012) 6566–6575.[15] S. Simic, Best possible bound for Jensens inequality, Applied Mathematics and Computation 215 (2009) 2224–2228.[16] E.L. Wachspress, A Rational Finite Element Basis, Vol. 114, Academic Press, New York, 1975.[17] T. Ju, S. Schaefer, J. Warren, M. Desbrun, A geometric construction of coordinates for convex polyhedra using polar duals, in: Proceedings of the

Symposium on Geometry Processing, 2005, pp. 181–186.[18] T. Ju, S. Schaefer, J. Warren, Mean value coordinates for closed triangular meshes, ACM Transactions on Graphics 24 (3) (2005) 561–566.[19] T. Ju, J. Warren, General Constructions of Barycentric Coordinates in a Convex Triangular Polyhedron. Tech. rep., Washington University in St. Louis,

2005.[20] J.L. Gout, Construction of a Hermite rational Wachspress type finite element, Computers and Mathematics with Applications 5 (1979) 337–347.[21] J.L. Gout, Interpolation error estimates on Hermite rational Wachspress type third degree finite element, Computers and Mathematics with

Applications 5 (4) (1979) 349–357.[22] M. Desbrun, M. Meyer, P. Alliez, Intrinsic parameterizations of surface meshes, Computer Graphics Forum 21 (2002) 209–218.[23] J. Schreiner, A. Asirvatham, E. Praun, H. Hoppe, Inter-surface mapping, ACM Transactions on Graphics 23 (3) (2004) 870–877.

1136 A. Guessab / Computers and Mathematics with Applications 66 (2013) 1120–1136

[24] T.W. Sederberg, S.R. Parry, Free-form deformation of solid geometric models, in: Computer Graphics, Proceedings of ACM SIGGRAPH, 1986,pp. 151–160.

[25] N. Sukumar, E.A. Malsch, Recent advances in the construction of polygonal interpolants, Archives of Computational Methods in Engineering 13 (1)(2006) 129–163.

[26] J.A. Kalman, Continuity and convexity of projections and barycentric coordinates in convex polyhedra, Pacific Journal of Mathematics, II (1961)1017–1022.

[27] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, 1979.[28] L. Chen, J. Xu, Optimal Delaunay triangulations, Journal of Computational Mathematics 22 (2004) 299–308.[29] M.S. Floater, K. Hormann, G. Kos, A general construction of barycentric coordinates over convex polygons, Advances in Computational Mathematics

24 (1) (2006) 311–331.[30] G.M. Ziegler, Lectures on Polytopes, Springer-Verlag, New York, 1995. 67, Elsevier Science, 2005.[31] A. Guessab, O. Nouisser, G. Schmeisser, Multivariate approximation by a combination of modified Taylor polynomials, Journal of Computational and

Applied Mathematics 196 (2006) 162–179.[32] M. Beck, S. Robins, Computing the Continuous Discretely: Integer-Point Enumeration in Polyhedra, Springer, 2007.[33] N. Sukumar, A. Tabarraei, Conforming polygonal finite elements, International Journal for Numerical Methods in Engineering 61 (12) (2004)

2045–2066.[34] P.-L. George, H. Borouchaki, Delaunay Triangulation and Meshing, Application to Finite Elements, Éditions Hermès, Paris, 1998. Translation from

French.[35] V.T. Rajan, Optimality of the Delaunay triangulation in Rd , Discrete & Computational Geometry 12 (1994) 189–202.[36] R.J. Renka, Algorithm 833: CSRFPACK-interpolation of scattered data with a C1 convexity-preserving surface, ACM Transactions on Mathematical

Software 30 (2) (2004) 200–211.[37] A. Li, Convexity preserving interpolation, Computer Aided Geometric Design 16 (1999) 127–147.