# Some Elementary Aspects of Means

Authors: Mowaffaq Hajja
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

We raise several elementary questions pertaining
to various aspects of means. These questions refer to both known and newly
introduced families of means, and include questions of characterizations
of certain families, relations among certain families, comparability among
the members of certain families, and concordance of certain sequences
of means. They also include questions about internality tests for certain
mean-looking functions and about certain triangle centers viewed as means
of the vertices. The questions are accessible to people with no background
in means, and it is also expected that these people can seriously investigate,
and contribute to the solutions of, these problems. The solutions
are expected to require no more than simple tools from analysis, algebra,
functional equations, and geometry.


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## Body

## 1. Definitions and Terminology

In all that follows,ℝ denotes the set of real numbers and 𝕁 denotes an interval in ℝ.By a data set (or a list) in a setS, we mean a finite subset of S in which repetition is allowed. Although the order in which the elements of a data set are written is not significant, we sometimes find it convenient to represent a data set in S of size n by a point in Sn, the cartesian product of n copies of S.We will call a data setA=(a1,…,an) in ℝ  ordered if a1≤⋯≤an. Clearly, every data set in ℝ may be assumed ordered.Amean of k variables (or a k-dimensional mean) on 𝕁 is defined to be any function ℳ:𝕁k→𝕁 that has the internality property
(1)min{a1,…,ak}≤ℳ(a1,…,ak)≤max{a1,…,ak}
for all aj in 𝕁. It follows that a mean ℳ must have the property ℳ(a,…,a)=a for all a in 𝕁.Most means that we encounter in the literature, and all means considered below, are also symmetric in the sense that(2)ℳ(a1,…,ak)=ℳ(aσ(1),…,aσ(n))
for all permutations σ on {1,…,n}, and 1-homogeneous in the sense that
(3)ℳ(λa1,…,λak)=λℳ(aσ(1),…,aσ(n))
for all permissible λ∈ℝ.Ifℳ and 𝒩 are two k-dimensional means on 𝕁, then we say that ℳ≤𝒩 if ℳ(a1,…,ak)≤𝒩(a1,…,ak) for all aj∈𝕁. We say that ℳ<𝒩 if ℳ(a1,…,ak)<𝒩(a1,…,ak) for all aj∈𝕁 for which a1,…,ak are not all equal. This exception is natural since ℳ(a,…,a) and 𝒩(a,…,a) must be equal, with each being equal to a. We say that ℳ and 𝒩 are comparable if ℳ≤𝒩 or 𝒩≤ℳ.Adistance (or a distance function) on a set S is defined to be any function d:S×S→[0,∞) that is symmetric and positive definite, that is,
(4)d(a,b)=d(b,a),∀a,b∈S,d(a,b)=0⟺a=b.
Thus a metric is a distance that satisfies the triangle inequality
(5)d(a,b)+d(b,c)≥d(a,c),∀a,b,c∈S,
a condition that we find too restrictive for our purposes.


## 2. Examples of Means

Thearithmetic, geometric, and harmonic means of two positive numbers were known to the ancient Greeks; see [1, pp. 84–90]. They are usually denoted by 𝒜, 𝒢, and ℋ, respectively, and are defined, for a,b>0, by
(6)𝒜(a,b)=a+b2,𝒢(a,b)=ab,ℋ(a,b)=21/a+1/b=2aba+b.
The celebrated inequalities
(7)ℋ(a,b)<𝒢(a,b)<𝒜(a,b)∀a,b>0
were also known to the Greeks and can be depicted in the well-known figure that is usually attributed to Pappus and that appears in [2, p. 364]. Several other less well known means were also known to the ancient Greeks; see [1, pp. 84–90].The three means above, and their natural extensions to any numbern of variables, are members of a large two-parameter family of means, known now as the Gini means and defined by
(8)Gr,s(x1,…,xn)=(Nr(x1,…,xn)Ns(x1,…,xn))1/(r-s),
where Nj are the Newton polynomials defined by
(9)Nj(x1,…,xn)=∑k=1n‍xkj.
Means of the type Gr,r-1 are known as Lehmer’s means, and those of the type Gr,0 are known as Hölder or power means. Other means that have been studied extensively are the elementary symmetric polynomial and elementary symmetric polynomial ratio means defined by
(10)(σrCrn)1/r,σr/Crnσr-1/Cr-1n,
where σr is the rth elementary symmetric polynomial in n variables, and where
(11)Crn=(nr).
These are discussed in full detail in the encyclopedic work [3, Chapters  III and V].It is obvious that the power means𝒫r defined by
(12)𝒫r(a1,…,an)=Gr,0(a1,…,an)=(a1r+⋯+anrn)1/r
that correspond to the values r=-1 and r=1 are nothing but the harmonic and arithmetic means ℋ and 𝒜, respectively. It is also natural to set
(13)𝒫0(a1,…,an)=𝒢(a1,…,an)=(a1…an)1/n,since
(14)limr→0(a1r+⋯+anrn)1/r=(a1…an)1/n
for all a1,…,an>0.The inequalities (7) can be written as 𝒫-1<𝒫0<𝒫1. These inequalities hold for any number of variables and they follow from the more general fact that 𝒫r(a1,…,an), for fixed a1,…,an>0, is strictly increasing with r. Power means are studied thoroughly in [3, Chapter  III].


## 3. Mean-Producing Distances and Distance Means

It is natural to think of the mean of any list of points in any set to be the point that isclosest to that list. It is also natural to think of a point as closest to a list of points if the sum of its distances from these points is minimal. This mode of thinking associates means to distances.Ifd is a distance on S, and if A=(a1,…,an) is a data set in S, then a d-mean of A is defined to be any element of S at which the function
(15)f(x)=∑i=1n‍d(x,ai)
attains its minimum. It is conceivable that (15) attains its minimum at many points, or nowhere at all. However, we shall be mainly interested in distances d on 𝕁 for which (15) attains its minimum at a unique point xA that, furthermore, has the property
(16)min{a:a∈A}≤xA≤max{a:a∈A}
for every data set A. Such a distance is called a mean-producing or a mean-defining distance, and the point xA is called the d-mean of A or the mean ofA  arising from the distanced and will be denoted by μd(A). A mean ℳ is called a distance mean if it is of the form μd for some distance d.Problem Set 1.
(1-a) Characterize those distances on𝕁 that are mean-producing.
(1-b) Characterize those pairs of mean producing distances on𝕁 that produce the same mean.
(1-c) Characterize distance means.


## 4. Examples of Mean-Producing Distances

Ifd0 is the discrete metric defined on ℝ by
(17)d0(a,b)={1ifa≠b,0ifa=b,
then the function f(x) in (15) is nothing but the number of elements in the given data set A that are different from x, and therefore every element having maximum frequency in A minimizes (15) and is hence a d0-mean of A. Thus the discrete metric gives rise to what is referred to in statistics as “the” mode of A. Due to the nonuniqueness of the mode, the discrete metric is not a mean-producing distance.Similarly, the usual metricd=d1 defined on ℝ by
(18)d1(a,b)=|a-b|
is not a mean-producing distance. In fact, it is not very difficult to see that if A=(a1,…,an) is an ordered data set of even size n=2m, then any number in the closed interval [am,am+1] minimizes
(19)∑j=1n‍|x-aj|
and is therefore a d1-mean of A. Similarly, one can show that if A is of an odd size n=2m-1, then am is the unique d1-mean of A. Thus the usual metric on ℝ gives rise to what is referred to in statistics as “the” median of A.On the other hand, the distanced2 defined on ℝ by
(20)d2(a,b)=(a-b)2
is a mean-producing distance, although it is not a metric. In fact, it follows from simple derivative considerations that the function
(21)∑j=1n‍(x-aj)2attains its minimum at the unique point(22)x=1n(∑j=1n‍aj).Thusd2 is a mean-producing distance, and the corresponding mean is nothing but the arithmetic mean.It is noteworthy that the three distances that come to mind most naturally give rise to the three most commonly used “means” in statistics. In this respect, it is also worth mentioning that a fourth mean of statistics, the so-calledmidrange, will be encountered below as a very natural limiting distance mean.The distancesd1 and d2 (and in a sense, d0 also) are members of the family dp of distances defined by
(23)dp(a,b)=|a-b|p.
It is not difficult to see that if p>1, then dp is a mean-producing distance. In fact, if A=(a1,…,an) is a given data set, and if
(24)f(x)=∑j=1n‍|x-aj|p,then(25)f′′(x)=p(p-1)∑j=1n‍|x-aj|p-2≥0,
with equality if and only if a1=⋯=an=x. Thus f is convex and cannot attain its minimum at more than one point. That it attains its minimum follows from the continuity of f(x), the compactness of [a1,an], and the obvious fact that f(x) is increasing on [an,∞) and is decreasing on (-∞,a1]. If we denote the mean that dp defines by μp, then μp(A) is the unique zero of
(26)∑j=1n‍sign(x-aj)|x-aj|p-1,
where sign(t) is defined to be 1 if t is nonnegative and −1 otherwise.Note that no matter whatp>1 is, the two-dimensional mean μp arising from dp is the arithmetic mean. Thus when studying μp, we confine our attention to the case when the number k of variables is greater than two. For such k, it is impossible in general to compute μp(A) in closed form.Problem 2.
It would be interesting to investigate comparability among{μp:p>1}.
It is highly likely that no two meansμp are comparable.


## 5. Deviation and Sparseness

Ifd is a mean-producing distance on S, and if μd is the associated mean, then it is natural to define the d-deviation 𝒟d(A) of a data set A=(a1,…,an) by an expression like
(27)𝒟d(A)=μd{d(μd(A),ai):1≤i≤n}.
Thus if d is defined by
(28)d(x,y)=(x-y)2,
then μd is nothing but the arithmetic mean or ordinary averageμ defined by
(29)μ=μ(a1,…,an)=a1+⋯+ann,
and 𝒟d is the (squared) standard deviation σ(2) given by
(30)σ(2)(a1,…,an)=|a1-μ|2+⋯+|an-μ|2n.In a sense, this provides an answer to those who are puzzled and mystified by the choice of the exponent 2 (and not any other exponent) in the standard definition of the standard deviation given in the right-hand side of (30). In fact, distance means were devised by the author in an attempt to remove that mystery. Somehow, we are saying that the ordinary average μ and the standard deviation σ(2) must be taken or discarded together, being both associated with the same distance d given in (28). Since few people question the sensibility of the definition of μ given in (29), accepting the standard definition of the standard deviation given in (30) as is becomes a must.It is worth mentioning that choosing an exponent other than 2 in (30) would result in an essentially different notion of deviations. More precisely, if one defines σ(k) by
(31)σ(k)(a1,…,an)=|a1-μ|k+⋯+|an-μ|kn,thenσ(k) and σ(2) would of course be unequal, but more importantly, they would not be monotone with respect to each other, in the sense that there would exist data sets A and B with σ(2)(A)>σ(k)(B) and σ(2)(A)<σ(k)(B). Thus the choice of the exponent k in defining deviations is not as arbitrary as some may feel. On the other hand, it is (27) and not (31) that is the natural generalization of (30). This raises the following, expectedly hard, problem.Problem 3.
Letd be the distance defined by d(x,y)=|x-y|k, and let the associated deviation 𝒟d defined in (27) be denoted by 𝒟k. Is 𝒟k monotone with respect to 𝒟2 for any k≠2, in the sense that
(32)𝒟k(A)>𝒟k(B)⟹𝒟2(A)>𝒟2(B)?We end this section by introducing the notion ofsparseness and by observing its relation with deviation. If d is a mean-producing distance on 𝕁, and if μd is the associated mean, then the d-sparseness𝒮d(A) of a data set A=(a1,…,an) in 𝕁 can be defined by
(33)𝒮d(A)=μd{d(ai,aj):1≤i<j≤n}.It is interesting that whend is defined by (28), the standard deviation coincides, up to a constant multiple, with the sparsenss. One wonders whether this pleasant property characterizes this distance d.Problem Set 4.
(4-a) Characterize those mean-producing distances whose associated mean is the arithmetic mean.
(4-b) Ifd is as defined in (28), and if d′ is another mean-producing distance whose associated mean is the arithmetic mean, does it follow that 𝒟d′ and 𝒟d are monotone with respect to each other?
(4-c) Characterize those mean-producing distancesδ for which the deviation 𝒟δ(A) is determined by the sparseness 𝒮δ(A) for every data set A, and vice versa.


## 6. Best Approximation Means

It is quite transparent that the discussion in the previous section regarding the distance meanμp, p>1, can be written in terms of best approximation in ℓpn, the vector space ℝn endowed with the p-norm ∥⋯∥p defined by
(34)∥(a1,…,an)∥p=(∑j=1n‍|aj|p)1/p.If we denote byΔ=Δn the line in ℝn consisting of the points (x1,…,xn) with x1=⋯=xn, then to say that a=μp(a1,…,an) is just another way of saying that the point (a,…,a) is a best approximant in Δn of the point (a1,…,an) with respect to the p-norm given in (34). Here, a point st in a subset S of a metric (or distance) space (T,D) is said to be a best approximant in S of t∈T if D(t,st)=min{D(t,s):s∈S}. Also, a subset S of (T,D) is said to be Chebyshev if every t in T has exactly one best approximant in S; see [4, p. 21].The discussion above motivates the following definition.Definition 1.
Let𝕁 be an interval in ℝ and let D be a distance on 𝕁n. If the diagonal Δ(𝕁n) of 𝕁n defined by
(35)Δ(𝕁n)={(a1,…,an)∈𝕁n:a1=⋯=an}
is Chebyshev (with respect toD), then the n-dimensional mean MD on 𝕁 defined by declaring MD(a1,…,an)=a if and only if (a,…,a) is the best approximant of (a1,…,an) in Δ(𝕁n) is called the Chebyshev or best approximationD-mean or the best approximation mean arising fromD.In particular, if one denotes byMp the best approximation n-dimensional mean on ℝ arising from (the distance on ℝn induced by) the norm ∥⋯∥p, then the discussion above says that Mp exists for all p>1 and that it is equal to μp defined in Section 4.In view of this, one may also defineM∞ to be the best approximation mean arising from the ∞-norm of ℓ∞n, that is, the norm ∥⋯∥∞ defined on ℝn by
(36)∥(a1,…,an)∥∞=max{|aj|:1≤j≤n}.It is not very difficult to see thatμ∞(A) is nothing but what is referred to in statistics as the mid-range of A. Thus if A=(a1,…,an) is an ordered data set, then
(37)M∞(A)=a1+an2.In view of the fact thatd∞ cannot be defined by anything like (23) and μ∞ is thus meaningless, natural question arises as to whether
(38)M∞(A)=limp→∞μp(A)(or equivalently=limp→∞Mp(A))for everyA. An affirmative answer is established in [5, Theorem  1]. In that theorem, it is also established that
(39)limp→qμp(A)(or  equivalentlylimp→qMp(A))=Mq(A)
for all q and all A. All of this can be expressed by saying that μp is continuous in p for p∈(1,∞] for all A.We remark that there is noobvious reason why (38) should immediately follow from the well known fact that
(40)limp→∞∥A∥p=∥A∥∞for all pointsA in ℝn.Problem Set 5.
Suppose thatδp is a sequence of distances on a set S that converges to a distance δ∞ (in the sense that limp→∞δp(a,b)=δ∞(a,b) for all a,  b in S). Let T⊆S. (5-a)
IfT is Chebyshev with respect to each δp, is it necessarily true that T is Chebyshev with respect to δ∞?(5-b)
IfT is Chebyshev with respect to each δp and with respect to δ∞ and if xp is the best approximant in T of x with respect to δp and x∞ is the best approximant in T of x with respect to δ∞, does it follow that xp converges to x∞?We end this section by remarking that ifM=Md is the n-dimensional best approximation mean arising from a distance d on 𝕁n, then d is significant only up to its values of the type d(u,v), where u∈Δ(𝕁n) and v∉Δ(𝕁n). Other values of d are not significant. This, together with the fact that
(41)every  mean  is  a  best  approximation  mean  arisingfrom  a  metric,
makes the study of best approximation means less interesting. Fact (41) was proved in an unduly complicated manner in [6], and in a trivial way based on a few-line set-theoretic argument in [7].Problem 6.
Given a meanℳ on 𝕁, a metric D on 𝕁 is constructed in [6] so that ℳ is the best approximation mean arising from D. Since the construction is extremely complicated in comparison with the construction in [7], it is desirable to examine the construction of D in [6] and see what other nice properties (such as continuity with respect to the usual metric) D has. This would restore merit to the construction in [6] and to the proofs therein and provide raison d’être for the so-called generalized means introduced there.


## 7. Towards a Unique Median

As mentioned earlier, the distanced1 on ℝ defined by (23) does not give rise to a (distance) mean. Equivalently, the 1-norm ∥⋯∥1 on ℝn defined by (34) does not give rise to a (best approximation) mean. These give rise, instead, to the many-valued function known as the median. Thus, following the statistician’s mode of thinking, one may set
(42)μ1(A)=M1(A)=the  median  interval  ofAμ1(A)=the  set  of  all  medians  ofA.
From a mathematician’s point of view, however, this leaves a lot to be desired, to say the least. The feasibility and naturality of defining μ∞ as the limit of μp as p approaches ∞ gives us a clue on how the median μ1 may be defined. It is a pleasant fact, proved in [5, Theorem  4], that the limit of μp(A) (equivalently of Mp(A)) as p decreases to 1 exists for every A∈ℝn and equals one of the medians described in (42). This limit can certainly be used as the definition of the median.Problem Set 7.
Letμp be as defined in Section 4, and let μ* be the limit of μp as p decreases to 1. (7-a)
Explore how the value ofμ*(A) compares with the common practice of taking the median of A to be the midpoint of the median interval (defined in (42) for various values of A.(7-b)
Isμ* continuous on ℝn? If not, what are its points of discontinuity?(7-c)
GivenA∈ℝn, is the convergence of μp(A) (as p decreases to 1) to μ*(A) monotone?The convergence ofμp(A) (as p decreases to 1) to μ*(A) is described in [5, Theorem  4], where it is proved that the convergence is ultimately monotone. It is also proved in [5, Theorem  5] that when n=3, then the convergence is monotone.It is of course legitimate to question the usefulness of defining the median to beμ*, but that can be left to statisticians and workers in relevant disciplines to decide. It is also legitimate to question the path that we have taken the limit along. In other words, it is conceivable that there exists, in addition to dp, a sequence dp′ of distances on ℝ that converges to d1 such that the limit μ**, as p decreases to 1, of their associated distance means μp′ is not the same as the limit μ* of μp. In this case, μ** would have as valid a claim as μ* to being the median. However, the naturality of dp may help accepting μ* as a most legitimate median.Problem Set 8.
Suppose thatδp and δp′, p∈ℕ, are sequences of distances on a set S that converge to the distances δ∞ and δ∞′, respectively (in the sense that limp→∞δp(a,b)=δ∞(a,b) for all a,b in S, etc.). (8-a)
If eachδp, p∈ℕ, is mean producing with corresponding mean mp, does it follow that δ∞ is mean producing? If so, and if the mean produced by δ∞ is m∞, is it necessarily true that mp converges to m∞?(8-b)
Ifδp and δp′, p∈ℕ∪{∞}, are mean producing distances with corresponding means mp and mp′, and if mp=mp′ for all p∈ℕ, does it follow that m∞=m∞′?


## 8. Examples of Distance Means

It is clear that the arithmetic mean is the distance mean arising from the the distanced2 given by d2(a,b)=(a-b)2. Similarly, the geometric mean on the set of positive numbers is the distance mean arising from the distance d𝒢 given by
(43)d𝒢(a,b)=(lna-lnb)2.In fact, this should not be amazing since the arithmetic mean𝒜 on ℝ and the geometric mean 𝒢 on (0,∞) are equivalent in the sense that there is a bijection g:(0,∞)→ℝ, namely g(x)=lnx, for which 𝒢(a,b)=g-1𝒜(g(a),g(b)) for all a,  b. Similarly, the harmonic and arithmetic means on (0,∞) are equivalent via the bijection h(x)=1/x, and therefore the harmonic mean is the distance mean arising from the distance dℋ given by
(44)dℋ(a,b)=(1a-1b)2.The analogous question pertaining to the logarithmic meanℒ defined by
(45)ℒ(a,b)=a-blna-lnb,a,b>0,remains open.Problem 9.
Decide whether the meanℒ (defined in (45)) is a distance mean.


## 9. Quasi-Arithmetic Means

Ak-dimensional mean ℳ on 𝕁 is called a quasi-arithmetic mean if there is a continuous strictly monotone function g from 𝕁 to an interval 𝕀 in ℝ such that
(46)ℳ(a1,…,ak)=g-1(𝒜(g(a1),…,g(ak)))for allaj in 𝕁. We have seen that the geometric and harmonic means are quasi-arithmetic and concluded that they are distance means. To see that ℒ is not quasi-arithmetic, we observe that the (two-dimensional) arithmetic mean, and hence any quasi-arithmetic mean ℳ, satisfies the elegant functional equation
(47)ℳ(ℳ(ℳ(a,b),b),ℳ(ℳ(a,b),a))=ℳ(a,b)
for all a,b>0. However, a quick experimentation with a random pair (a,b) shows that (47) is not satisfied by ℒ.This shows thatℒ is not quasi-arithmetic, but does not tell us whether ℒ is a distance mean, and hence does not answer Problem 9.The functional equation (47) is a weaker form of the functional equation
(48)ℳ(ℳ(a,b),ℳ(c,d))=ℳ(ℳ(a,c),ℳ(b,d))for alla,b,c,d>0. This condition, together with the assumption that ℳ is strictly increasing in each variable, characterizes two-dimensional quasi-arithmetic means; see [8, Theorem  1, pp. 287–291]. A thorough discussion of quasi-arithmetic means can be found in [3, 8].Problem 10.
Decide whether a meanℳ that satisfies the functional equation (47) (together with any necessary smoothness conditions) is necessarily a quasi-arithmetic mean.


## 10. Deviation Means

Deviation means were introduced in [9] and were further investigated in [10]. They are defined as follows.A real-valued functionE=E(x,t) on ℝ2 is called a deviation if E(x,x)=0 for all x and if E(x,t) is a strictly decreasing continuous function of t for every x. If E is a deviation, and if x1,…,xn are given, then the E-deviation mean of x1,…,xn is defined to be the unique zero of
(49)E(x1,t)+⋯+E(xn,t).It is direct to see that (49) has a unique zero and that this zero does indeed define a mean.Problem 11.
Characterize deviation means and explore their exact relationship with distance means.
IfE is a deviation, then (following [11]), one may define dE by
(50)dE(x,t)=∫xt‍E(x,s)ds.
ThendE(x,t)≥0 and dE(x,t) is a strictly convex function in t for every x. The E-deviation mean of x1,…,xn is nothing but the unique value of t at which dE(x1,t)+⋯+dE(xn,t) attains its minimum. Thus if dE happens to be symmetric, then dE would be a distance and the E-deviation mean would be the distance mean arising from the distance dE.


## 11. Other Ways of Generating New Means

Iff and g are differentiable on an open interval 𝕁, and if a<b are points in 𝕁 such that f(b)≠f(a), then there exists, by Cauchy’s mean value theorem, a point c in (a,b), such that
(51)f′(c)g′(c)=g(b)-g(a)f(b)-f(a).Iff and g are such that c is unique for every a,  b, then we call c the Cauchy mean of a and b corresponding to the functions f and g, and we denote it by 𝒞f,g(a,b).Another natural way of defining means is to take a continuous functionF that is strictly monotone on 𝕁, and to define the mean of a,b∈𝕁, a≠b, to be the unique point c in (a,b) such that
(52)F(c)=1b-a∫ab‍F(x)dx.We callc the mean value (mean) of a and b corresponding to F, and we denote it by 𝒱(a,b).Clearly, ifH is an antiderivative of F, then (53) can be written as
(53)H′(c)=H(b)-H(a)b-a.Thus𝒱F(a,b)=𝒞H,E(a,b), where E is the identity function.For more on the these two families of means, the reader is referred to [12] and [13], and to the references therein.In contrast to the attitude of thinking of the mean as the number that minimizes a certain function, there is what one may call the Chisini attitude that we now describe. A functionf on 𝕁n may be called a Chisini function if and only if the equation
(54)f(a1,…,an)=f(x,…,x)has a unique solutionx=a∈[a1,an] for every ordered data set (a1,…,an) in 𝕁. This unique solution is called the Chisini mean associated to f. In Chisini’s own words, xis said to be the mean ofnnumbersx1,…,xnwith respect to a problem, in which a function of themf(x1,…,xn)is of interest, if the function assumes the same value when all thexhare replaced by the mean valuex:f(x1,…,xn)=f(x,…,x); see [14, page 256] and [1]. Examples of such Chisini means that arise in geometric configurations can be found in [15].Problem 12.
Investigate how the families of distance, deviation, Cauchy, mean value, and Chisini means are related.


## 12. Internality Tests

According to the definition of a mean, all that is required of a functionℳ:𝕁n→𝕁 to be a mean is to satisfy the internality property
(55)min{a1,…,ak}≤ℳ(a1,…,ak)≤max{a1,…,ak}for allaj∈𝕁. However, one may ask whether it is sufficient, for certain types of functions ℳ, to verify (55) for a finite, preferably small, number of well-chosen n-tuples. This question is inspired by certain elegant theorems in the theory of copositive forms that we summarize below.


### 12.1. Copositivity Tests for Quadratic and Cubic Forms

By a (real)form in n variables, we shall always mean a homogeneous polynomial F=F(x1,…,xn) in the indeterminates x1,…,xn having coefficients in ℝ. When the degree t of a form F is to be emphasized, we call F a t-form. Forms of degrees 1, 2, 3, 4, and 5 are referred to as linear, quadratic, cubic, quartic, and quintic forms, respectively.The set of allt-forms in n variables is a vector space (over ℝ) that we shall denote by 𝔽t(n). It may turn out to be an interesting exercise to prove that the set
(56){∏j=1d‍Njej:∑j=1d‍jej=d}is a basis, whereNj is the Newton polynomial defined by
(57)Nj=∑k=1n‍xkj.The statement above is quite easy to prove in the special cased≤3, and this is the case we are interested in in this paper. We also discard the trivial case n=1  and assume always that n≥2.Linear forms can be written asaN1, and they are not worth much investigation. Quadratic forms can be written as
(58)Q=aN12+bN2=a(∑k=1n‍xk)2+b(∑k=1n‍xk2).Cubic and quartic forms can be written, respectively, as(59)aN13+bN1N2+cN3,aN14+bN12N2+cN1N3+dN22.A formF=F(x1,…,xn) is said to be copositive if f(a1,…,an)≥0 for all xi≥0. Copositive forms arise in the theory of inequalities and are studied in [14] (and in references therein). One of the interesting questions that one may ask about forms pertains to algorithms for deciding whether a given form is copositive. This problem, in full generality, is still open. However, for quadratic and cubic forms, we have the following satisfactory answers.Theorem 2.
LetF=F(x1,…,xn) be a real symmetric form in any number n≥2 of variables. Let vm(n), 1≤m≤n, be the n-tuple whose first m coordinates are 1’s and whose remaining coordinates are 0′s. (i)
IfF is quadratic, then F is copositive if and only if F≥0 at the two test n-tuples
(60)v1(n)=(1,0,…,0),vn(n)=(1,1,…,1).(ii)
IfF is cubic, then F is copositive if and only if F≥0 at the n test n-tuples
(61)vm(n)=(1,…,1︷m,0,…,0︷n-m),1≤m≤n.Part (i) is a restatement of Theorem  1(a) in [16]. Theorem  1(b) there is related and can be restated as
(62)F(a1,…,an)≥0,∀ai∈ℝ,⟺F≥0at  the3n-tuples(1,0,…,0),(1,1,…,1),(1,-1,0,…,0).Part (ii) was proved in [17] for n≤3 and in [18] for all n. Two very short and elementary inductive proofs are given in [19].It is worth mentioning that then test n-tuples in (61) do not suffice for establishing the copositivity of a quartic form even when n=3. An example illustrating this that uses methods from [20] can be found in [19]. However, an algorithm for deciding whether a symmetric quartic form f in n variables is copositive that consists in testing f at n-tuples of the type
(63)(a,…,a︷m,1,…,1︷r,0,…,0︷n-m-r),0≤m,r≤n,m+r≤nis established in [21]. It is also proved there that if n=3, then the same algorithm works for quintics but does not work for forms of higher degrees.


### 12.2. Internality Tests for Means Arising from Symmetric Forms

Let𝔽t(n) be the vector space of all real t-forms in n variables, and let Nj, 1≤j≤d, be the Newton polynomials defined in (57). Means of the type
(64)ℳ=(FrFs)1/(r-s),whereFj is a symmetric form of degree j, are clearly symmetric and 1-homogeneous, and they abound in the literature. These include the family of Gini means Gr,s defined in (8) (and hence the Lehmer and Hölder means). They also include the elementary symmetric polynomial and elementary symmetric polynomial ratio means defined earlier in (10).In view of Theorem2 of the previous section, it is tempting to ask whether the internality of a function ℳ of the type described in (64) can be established by testing it at a finite set of test n-tuples. Positive answers for some special cases of (64), and for other related types, are given in the following theorem.Theorem 3.
LetL, Q, and C be real symmetric forms of degrees 1, 2, and 3, respectively, in any number n≥2 of nonnegative variables. Let vk(n), 1≤k≤n, be as defined in Theorem 2. (i)
Q is internal if and only if it is internal at the two test n-tuples: vn(n)=(1,1,…,1) and vn-1(n)=(1,1,…,1,0).(ii)
Q/L is internal if and only if it is internal at the two test n-tuples: vn(n)=(1,1,…,1) and v1(n)=(1,0,…,0).(iii)
Ifn≤4, then C3 is internal if and only if it is internal at the n test n-tuples
(65)vm(n)=(1,…,1︷m,0,…,0︷n-m),1≤m≤n.Parts (i) and (ii) are restatements of Theorems  3 and  5 in [16]. Part (iii) is proved in [22] in a manner that leaves a lot to be desired. Besides being rather clumsy, the proof works for n≤4 only. The problem for n≥5, together with other open problems, is listed in the next problem set.Problem Set 13.
LetL, Q, and C be real symmetric cubic forms of degrees 1, 2, and 3, respectively, in n non-negative variables. (13-a)
Prove or disprove thatC3 is internal if and only if it is internal at the n test n-tuples
(66)vm(n)=(1,…,1︷m,0,…,0︷n-m),1≤m≤n.(13-b)
Find, or prove the nonexistence of, a finite setT of test n-tuples such that the internality of C/Q at the n-tuples in T gurantees its internality at all nonnegative n-tuples.(13-c)
Find, or prove the nonexistence of, a finite setT of test n-tuples such that the internality of L±Q at the n-tuples in T guarantees its internality at all non-negative n-tuples.Problem (13-b) is open even forn=2. In Section  6 of [15], it is shown that the two pairs (1,0) and (1,1) do not suffice as test pairs.As for Problem (13-c), we refer the reader to [23], where means of the type L±Q were considered. It is proved in Theorem 2 there that when Q has the special form a∏1≤i<j≤n‍(xi-xj)2, then L±Q is internal if and only if it is internal at the two test n-tuples vn(n)=(1,1,…,1) and vn-1(n)=(1,1,…,1,0). In the general case, sufficient and necessary conditions for internality of L±Q, in terms of the coefficients of L and Q, are found in [23, Theorem  3]. However, it is not obvious whether these conditions can be rewritten in terms of test n-tuples in the manner done in Theorem 3.


## 12.1. Copositivity Tests for Quadratic and Cubic Forms

By a (real)form in n variables, we shall always mean a homogeneous polynomial F=F(x1,…,xn) in the indeterminates x1,…,xn having coefficients in ℝ. When the degree t of a form F is to be emphasized, we call F a t-form. Forms of degrees 1, 2, 3, 4, and 5 are referred to as linear, quadratic, cubic, quartic, and quintic forms, respectively.The set of allt-forms in n variables is a vector space (over ℝ) that we shall denote by 𝔽t(n). It may turn out to be an interesting exercise to prove that the set
(56){∏j=1d‍Njej:∑j=1d‍jej=d}is a basis, whereNj is the Newton polynomial defined by
(57)Nj=∑k=1n‍xkj.The statement above is quite easy to prove in the special cased≤3, and this is the case we are interested in in this paper. We also discard the trivial case n=1  and assume always that n≥2.Linear forms can be written asaN1, and they are not worth much investigation. Quadratic forms can be written as
(58)Q=aN12+bN2=a(∑k=1n‍xk)2+b(∑k=1n‍xk2).Cubic and quartic forms can be written, respectively, as(59)aN13+bN1N2+cN3,aN14+bN12N2+cN1N3+dN22.A formF=F(x1,…,xn) is said to be copositive if f(a1,…,an)≥0 for all xi≥0. Copositive forms arise in the theory of inequalities and are studied in [14] (and in references therein). One of the interesting questions that one may ask about forms pertains to algorithms for deciding whether a given form is copositive. This problem, in full generality, is still open. However, for quadratic and cubic forms, we have the following satisfactory answers.Theorem 2.
LetF=F(x1,…,xn) be a real symmetric form in any number n≥2 of variables. Let vm(n), 1≤m≤n, be the n-tuple whose first m coordinates are 1’s and whose remaining coordinates are 0′s. (i)
IfF is quadratic, then F is copositive if and only if F≥0 at the two test n-tuples
(60)v1(n)=(1,0,…,0),vn(n)=(1,1,…,1).(ii)
IfF is cubic, then F is copositive if and only if F≥0 at the n test n-tuples
(61)vm(n)=(1,…,1︷m,0,…,0︷n-m),1≤m≤n.Part (i) is a restatement of Theorem  1(a) in [16]. Theorem  1(b) there is related and can be restated as
(62)F(a1,…,an)≥0,∀ai∈ℝ,⟺F≥0at  the3n-tuples(1,0,…,0),(1,1,…,1),(1,-1,0,…,0).Part (ii) was proved in [17] for n≤3 and in [18] for all n. Two very short and elementary inductive proofs are given in [19].It is worth mentioning that then test n-tuples in (61) do not suffice for establishing the copositivity of a quartic form even when n=3. An example illustrating this that uses methods from [20] can be found in [19]. However, an algorithm for deciding whether a symmetric quartic form f in n variables is copositive that consists in testing f at n-tuples of the type
(63)(a,…,a︷m,1,…,1︷r,0,…,0︷n-m-r),0≤m,r≤n,m+r≤nis established in [21]. It is also proved there that if n=3, then the same algorithm works for quintics but does not work for forms of higher degrees.


## 12.2. Internality Tests for Means Arising from Symmetric Forms

Let𝔽t(n) be the vector space of all real t-forms in n variables, and let Nj, 1≤j≤d, be the Newton polynomials defined in (57). Means of the type
(64)ℳ=(FrFs)1/(r-s),whereFj is a symmetric form of degree j, are clearly symmetric and 1-homogeneous, and they abound in the literature. These include the family of Gini means Gr,s defined in (8) (and hence the Lehmer and Hölder means). They also include the elementary symmetric polynomial and elementary symmetric polynomial ratio means defined earlier in (10).In view of Theorem2 of the previous section, it is tempting to ask whether the internality of a function ℳ of the type described in (64) can be established by testing it at a finite set of test n-tuples. Positive answers for some special cases of (64), and for other related types, are given in the following theorem.Theorem 3.
LetL, Q, and C be real symmetric forms of degrees 1, 2, and 3, respectively, in any number n≥2 of nonnegative variables. Let vk(n), 1≤k≤n, be as defined in Theorem 2. (i)
Q is internal if and only if it is internal at the two test n-tuples: vn(n)=(1,1,…,1) and vn-1(n)=(1,1,…,1,0).(ii)
Q/L is internal if and only if it is internal at the two test n-tuples: vn(n)=(1,1,…,1) and v1(n)=(1,0,…,0).(iii)
Ifn≤4, then C3 is internal if and only if it is internal at the n test n-tuples
(65)vm(n)=(1,…,1︷m,0,…,0︷n-m),1≤m≤n.Parts (i) and (ii) are restatements of Theorems  3 and  5 in [16]. Part (iii) is proved in [22] in a manner that leaves a lot to be desired. Besides being rather clumsy, the proof works for n≤4 only. The problem for n≥5, together with other open problems, is listed in the next problem set.Problem Set 13.
LetL, Q, and C be real symmetric cubic forms of degrees 1, 2, and 3, respectively, in n non-negative variables. (13-a)
Prove or disprove thatC3 is internal if and only if it is internal at the n test n-tuples
(66)vm(n)=(1,…,1︷m,0,…,0︷n-m),1≤m≤n.(13-b)
Find, or prove the nonexistence of, a finite setT of test n-tuples such that the internality of C/Q at the n-tuples in T gurantees its internality at all nonnegative n-tuples.(13-c)
Find, or prove the nonexistence of, a finite setT of test n-tuples such that the internality of L±Q at the n-tuples in T guarantees its internality at all non-negative n-tuples.Problem (13-b) is open even forn=2. In Section  6 of [15], it is shown that the two pairs (1,0) and (1,1) do not suffice as test pairs.As for Problem (13-c), we refer the reader to [23], where means of the type L±Q were considered. It is proved in Theorem 2 there that when Q has the special form a∏1≤i<j≤n‍(xi-xj)2, then L±Q is internal if and only if it is internal at the two test n-tuples vn(n)=(1,1,…,1) and vn-1(n)=(1,1,…,1,0). In the general case, sufficient and necessary conditions for internality of L±Q, in terms of the coefficients of L and Q, are found in [23, Theorem  3]. However, it is not obvious whether these conditions can be rewritten in terms of test n-tuples in the manner done in Theorem 3.


## 13. Extension of Means, Concordance of Means

The two-dimensional arithmetic mean𝒜(2) defined by
(67)𝒜(2)(a1,a2)=a1+a22can be extended to any dimensionk by setting
(68)𝒜(k)(a1,…,ak)=a1+⋯+akk.Although very few people would disagree on this, nobody can possibly give a mathematically sound justification of the feeling that the definition in (68) is the only (or even the best) definition that makes the sequence A(k) of means harmonious or concordant. This does not seem to be an acceptable definition of the notion of concordance.In a private communication several years ago, Professor Zsolt Páles told me that Kolmogorov suggested calling a sequenceℳ(k) of means on 𝕁, where ℳ(k) is k-dimensional, concordant if for every m and n and every ai,  bi in 𝕁, we have
(69)ℳ(n+m)(a1,…,an,b1,…,bm)=ℳ(2)(ℳ(n)(a1,…,an),ℳm(b1,…,bm)).He also told me that such a definition is too restrictive and seems to confirm concordance in the case of the quasi-arithmetic means only.Problem 14.
Suggest a definition of concordance, and test it on sequences of means that you feel concordant. In particular, test it on the existing generalizations, to higher dimensions, of the logarithmic meanℒ defined in (45).


## 14. Distance Functions in Topology

Distance functions, which are not necessarily metrics, have appeared early in the literature on topology. Given a distance functiond on any set X, one may define the open ballB(a,r) in the usual manner, and then one may declare a subset A⊆X  open if it contains, for every a∈A, an open ball B(a,r) with r>0. If d has the triangle inequality, then one can proceed in the usual manner to create a topology. However, for a general distance d, this need not be the case, and distances that give rise to a coherent topology in the usual manner are called semimetrics and they are investigated and characterized in [24–29]. Clearly, these are the distances d for which the family {B(a,r):r>0} of open balls centered at a∈S forms a local base at a for every a in X.


## 15. Centers and Center-Producing Distances

A distanced may be defined on any set S whatsoever. In particular, if d is a distance on ℝ2 and if the function f(X) defined by
(70)f(X)=∑i=1n‍d(X,Ai)attains its minimum at a unique pointX0 that lies in the convex hull of {A1,…,An} for every choice of A1,…,An in ℝ2, then d will be called a center-producing distance.The Euclidean metricd1 on ℝ2 produces the Fermat-Torricelli center. This is defined to be the point whose distances from the given points have a minimal sum. Its square, d2, which is just a distance but not a metric, produces the centroid. This is the center of mass of equal masses placed at the given points. It would be interesting to explore the centers defined by dp for other values of p.Problem 15.
Letdp, p>1, be the distance defined on ℝ2 by dp(A,B)=∥A-B∥p, and let ABC be a triangle. Let Zp=Zp(A,B,C) be the point that minimizes
(71)dp(Z,A)+dp(Z,B)+dp(Z,C)=∥Z-A∥p+∥Z-B∥p+∥Z-C∥p.Investigate how Zp, p≥1, are related to the known triangle centers, and study the curve traced by them.The papers [30, 31] may turn out to be relevant to this problem.



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*Source: 689560-2013-05-07.xml*