- Open Access
Convex combinations, barycenters and convex functions
© Pavić; licensee Springer 2013
- Received: 27 June 2012
- Accepted: 1 February 2013
- Published: 20 February 2013
The article first shows one alternative definition of convexity in the discrete case. The correlation between barycenters, Jensen’s inequality and convexity is studied in the integral case. The Hermite-Hadamard inequality is also obtained as a consequence of a concept of barycenters. Some derived results are applied to the quasi-arithmetic means and especially to the power means.
MSC:26A51, 26D15, 28A10, 28A25.
- convex combination
- convex function
- Jensen’s inequality
- quasi-arithmetic mean
Sets with the common barycenter are observed in geometry, mechanics dealing with mass densities and probability theory in the study of random variables. Development and application of the theory of convex functions also includes barycenters. The following result, expressed by the measure and integral, is the most commonly used.
holds for every μ-integrable convex function .’
The related problems with different types of measures and mathematical expectations were investigated in . The inequality in (1.2) under the condition in (1.1) was extended in . The intention of this paper is still more to connect the quoted implication (in the extended form) with convex functions, in the discrete and integral case. We also wanted to insert the quasi-arithmetic means into this implication.
The quoted result was actually observed in Banach spaces. So, it was assumed that A and B are bounded closed convex subsets of a Banach space E such that and is a convex function. The opposite examples are found in  already for and .
Throughout the whole paper, we suppose that is a non-degenerate interval. Subintervals from I will also be non-degenerate. Convex hull of a set X will be denoted by coX.
The main results of the paper are presented in Sections 2 and 3.
In this section, we show the connection between the convex combinations and the convex functions. The basic form of Jensen’s inequality is obtained using the assumption of the equality of convex combinations. An alternative definition of convexity is also presented.
Throughout the section, we will assume that n is a positive number greater than or equal to 2, i.e., .
An elementary mean of points is the arithmetic mean . A discrete generalization of the arithmetic mean is the convex combination or the weighted mean with coefficients such that .
holds for every convex function .
using and and by putting the sums instead of integrals. So, the implication of Theorem A can be proved without applying the basic Jensen inequality.
holds for every function which satisfies the implication of Theorem A.
The next consequence is the basic form of Jensen’s inequality, as the main result in this section.
holds for every function which satisfies the implication of Theorem A.
Proof Let with . Without loss of generality, suppose that all are pairwise different and all .
because . □
So, using Theorem A, we can derive the basic Jensen inequality. The previous results can be written in the following theorem as the alternative definition of convexity.
Theorem 2.3 A function is convex if and only if it satisfies the implication of Theorem A.
In this section, we show the connection between the convexity and the barycenters. The integral form of Jensen’s inequality for the measures which satisfy some conditions is obtained using the barycenters.
Integral generalizations of the concept of arithmetic mean in the finite measure spaces are the integral arithmetic mean or the barycenter of measurable set and the integral arithmetic mean of integrable function; see [, p.44]. In particular, if we have a probabilistic measure, then the integral arithmetic mean of a random variable is just its mathematical expectation.
Note that , where is an identity function on A. If A is the interval, then its μ-barycenter belongs to A. If A is the interval and f is continuous on A, then its μ-arithmetic mean on A belongs to .
holds for every convex μ-integrable function .
The version of Theorem B for the bounded closed intervals A and B was proved in [, Proposition 1] by using the chord line when . The proof was realized without applying the integral Jensen inequality. The same proof can be applied to Theorem B with and .
It is unfortunate that Theorem B is not valid for the convex functions of several variables. Such examples for the convex function of two and three variables are shown in [, Example 1,2].
The next corollary is the generalization of Theorem B. It can be also useful in some applications, especially in applications on quasi-arithmetic means.
holds for every convex function provided that is μ-integrable.
The following is the integral analogy of Corollary 2.1.
holds for every μ-integrable function which satisfies the implication of Theorem B.
The concept of barycenter enables the realization of the most important inequalities such as the Jensen inequality and the Hermite-Hadamard inequality. This approach requires fine measures.
are continuous and monotone on . If additionally the measure μ is positive on the intervals from I, then the above functions are strictly monotone.
In the rest of this section, we will use the continuous finite measure on I which is positive on the intervals from I, that is, for every interval .
Lemma 3.3 Let μ be a continuous finite measure on I which is positive on the intervals from I.
Proof Take a point a from the interior of I.
Since g is continuous, and , there must be a point such that . In this case we can take . If , then we increase until we obtain one of the previous two cases.
and repeat the previous procedure to determine . □
is strictly decreasing continuous on with .
The following consequence is the integral form of Jensen’s inequality, as the main result in this section.
Theorem 3.5 Let μ be a continuous finite measure on I which is positive on the intervals from I.
holds for every continuous μ-integrable function which satisfies the implication of Theorem B for unions B of intervals from I and bounded intervals .
be its μ-barycenter. We observe three cases depending on the μ-barycenter a.
which ends the proof of this case.
If a is the boundary point of B, then we take small and put or . It provides that μ-barycenter of belongs to the interior of . First, we apply the above procedure to and its μ-barycenter , and after that allow .
If a does not belong to B and if a is not the boundary point of B, then we take small and put . It provides that μ-barycenter of belongs to the interior of . We apply the procedure from the first case to , and after that let . □
where the series of the μ-barycenters of intervals converges to a.
Thus, using Theorem B, we can realize the integral form of Jensen’s inequality for continuous functions, unions of intervals and continuous finite measures which are positive on intervals. The following is the equivalent connection between convexity and Theorem B.
Theorem 3.7 Let μ be a continuous finite measure on I which is positive on the intervals from I. A continuous μ-integrable function is convex if and only if it satisfies the implication of Theorem B for finite unions B of intervals from I and bounded intervals .
Proof The necessity follows from Theorem B. Let us prove the sufficiency on the interior of I. Take any convex combination of two different points x and y from the interior of I with the positive coefficients p and q. Suppose .
The basic idea of the proof is to determine the small intervals and with the barycenters x and y such that . Suppose we have and with barycenters x and y. If , then we decrease . If , then we decrease .
which ends the proof. □
For details on global bounds for generalized Jensen’s inequality, see .
The Hermite-Hadamard inequality is also the consequence of Theorem B.
Corollary 3.8 Let μ be a continuous finite measure on I which is positive on the intervals from I.
holds for every continuous function which satisfies the implication of Theorem B for finite unions B of intervals from I and bounded intervals .
We construct an interval by increasing the interval , after we have constructed intervals and by decreasing the intervals and .
Letting , we obtain the inequality in (3.10). □
An interesting version of the Hermite-Hadamard inequality in a non-positive curvature space was obtained in .
In the applications of convexity, we often use strictly monotone continuous functions such that ψ is convex with respect to φ (ψ is φ-convex), that is, is convex by [, Definition 1.19]. A similar notation is used for concavity.
which belongs to I because belongs to .
If ψ is either φ-convex and decreasing or φ-concave and increasing, then the reverse double inequality is valid in (4.3).
If A is the interval, then its φ-quasi-arithmetic mean belongs to A because the point belongs to . If A is not connected, then may be outside of A.
If ψ is either φ-convex and decreasing or φ-concave and increasing, then the reverse double inequality is valid in (4.6).
Finally, we apply the increasing function to the above inequalities and get the double inequality in (4.6). □
Respecting the mark for integral power mean, it comes next .
holds for .
Proof The proof of corollary follows from Theorem 4.1 with functions and for or for . □
General forms and refinements of quasi-arithmetic means can be found in .
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