Some Elementary Aspects of Means

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Volume 2013, Article ID 689560,9pages

Research Article

Some Elementary Aspects of Means

Mowaffaq Hajja

Department of Mathematics, Yarmouk University, Irbid, Jordan

Correspondence should be addressed to Mowaffaq Hajja;

Received 24 December 2012; Accepted 31 March 2013

Academic Editor: Peter Bullen

Copyright © 2013 Mowaffaq Hajja. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Definitions and Terminology

In all that follows,Rdenotes the set of real numbers andJ

denotes an interval inR.

By a data set (or a list) in a set𝑆, we mean a finite subset of

𝑆in 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 in𝑆of

size𝑛by a point in𝑆𝑛, the cartesian product of𝑛copies of𝑆.

We will call a data set𝐴 = (ğ‘Ž1, . . . , ğ‘Žğ‘›)inRordered if

ğ‘Ž1 ≤ ⋅ ⋅ ⋅ ≤ ğ‘Žğ‘›. Clearly, every data set inRmay be assumed ordered.

Ameanof𝑘variables (or a𝑘-dimensional mean) onJis

defined to be any functionM:J𝑘 → Jthat has the


min{ğ‘Ž1, . . . , ğ‘Žğ‘˜} ≤M(ğ‘Ž1, . . . , ğ‘Žğ‘˜) ≤max{ğ‘Ž1, . . . , ğ‘Žğ‘˜} (1)

for allğ‘Žğ‘—inJ. It follows that a meanMmust have the property

M(ğ‘Ž, . . . , ğ‘Ž) = ğ‘Žfor allğ‘ŽinJ.

Most means that we encounter in the literature, and all means considered below, are also symmetric in the sense that

M(ğ‘Ž1, . . . , ğ‘Žğ‘˜) =M(ğ‘ŽğœŽ(1), . . . , ğ‘ŽğœŽ(𝑛)) (2)

for all permutationsğœŽon{1, . . . , 𝑛}, and1-homogeneousin the

sense that

M(ğœ†ğ‘Ž1, . . . , ğœ†ğ‘Žğ‘˜) = 𝜆M(ğ‘ŽğœŽ(1), . . . , ğ‘ŽğœŽ(𝑛)) (3)

for all permissible𝜆 ∈R.

IfMandNare two𝑘-dimensional means onJ, then we

say thatM ≤ N ifM(ğ‘Ž1, . . . , ğ‘Žğ‘˜) ≤ N(ğ‘Ž1, . . . , ğ‘Žğ‘˜)for all

ğ‘Žğ‘— ∈J. We say thatM<NifM(ğ‘Ž1, . . . , ğ‘Žğ‘˜) <N(ğ‘Ž1, . . . , ğ‘Žğ‘˜)

for all ğ‘Žğ‘— ∈ J for whichğ‘Ž1, . . . , ğ‘Žğ‘˜ are not all equal. This

exception is natural sinceM(ğ‘Ž, . . . , ğ‘Ž)andN(ğ‘Ž, . . . , ğ‘Ž)must

be equal, with each being equal toğ‘Ž. We say thatMandN


Adistance(ora distance function) on a set𝑆is defined to

be any function𝑑 : 𝑆 × 𝑆 → [0, ∞)that issymmetricand

positive definite, that is,

𝑑 (ğ‘Ž, 𝑏) = 𝑑 (𝑏, ğ‘Ž) , âˆ€ğ‘Ž, 𝑏 ∈ 𝑆,

𝑑 (ğ‘Ž, 𝑏) = 0 ⇐⇒ ğ‘Ž = 𝑏. (4)

Thus ametricis a distance that satisfies the triangle inequality

𝑑 (ğ‘Ž, 𝑏) + 𝑑 (𝑏, 𝑐) ≥ 𝑑 (ğ‘Ž, 𝑐) , âˆ€ğ‘Ž, 𝑏, 𝑐 ∈ 𝑆, (5)

a condition that we find too restrictive for our purposes.

2. Examples of Means

Thearithmetic,geometric, andharmonicmeans of two


pp. 84–90]. They are usually denoted by A, G, and H,

respectively, and are defined, forğ‘Ž, 𝑏 > 0, by

A(ğ‘Ž, 𝑏) = ğ‘Ž + 𝑏 2 , G(ğ‘Ž, 𝑏) = âˆšğ‘Žğ‘,

H(ğ‘Ž, 𝑏) =1/ğ‘Ž + 1/𝑏2 = ğ‘Ž + 𝑏2ğ‘Žğ‘.


The celebrated inequalities

H(ğ‘Ž, 𝑏) <G(ğ‘Ž, 𝑏) <A(ğ‘Ž, 𝑏) âˆ€ğ‘Ž, 𝑏 > 0 (7)

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–


The three means above, and their natural extensions to

any number 𝑛 of variables, are members of a large

two-parameter family of means, known now as theGinimeans and

defined by

𝐺𝑟,𝑠(𝑥1, . . . , 𝑥𝑛) = (𝑁𝑟(𝑥1, . . . , 𝑥𝑛) 𝑁𝑠(𝑥1, . . . , 𝑥𝑛))


, (8)

where𝑁𝑗are the Newton polynomials defined by

𝑁𝑗(𝑥1, . . . , 𝑥𝑛) =∑𝑛


𝑥𝑗𝑘. (9)

Means of the type𝐺𝑟,𝑟−1are known asLehmer’s means, and

those of the type𝐺𝑟,0are known asH¨olderorpowermeans.

Other means that have been studied extensively are the

elementary symmetric polynomialandelementary symmetric polynomial ratiomeans defined by


𝐶𝑛 𝑟)


, ğœŽğ‘Ÿ/𝐶𝑛𝑟



whereğœŽğ‘Ÿ is the 𝑟th elementary symmetric polynomial in𝑛

variables, and where

𝐶𝑛𝑟 = (𝑛𝑟). (11)

These are discussed in full detail in the encyclopedic work [3,

Chapters III and V].

It is obvious that the power meansP𝑟defined by

P𝑟(ğ‘Ž1, . . . , ğ‘Žğ‘›) = 𝐺𝑟,0(ğ‘Ž1, . . . , ğ‘Žğ‘›) = (ğ‘Ž1𝑟+ ⋅ ⋅ ⋅ + ğ‘Žğ‘Ÿğ‘›

𝑛 )



that correspond to the values𝑟 = −1and𝑟 = 1are nothing but

the harmonic and arithmetic meansHandA, respectively. It

is also natural to set

P0(ğ‘Ž1, . . . , ğ‘Žğ‘›) =G(ğ‘Ž1, . . . , ğ‘Žğ‘›) = (ğ‘Ž1. . . ğ‘Žğ‘›)1/𝑛, (13)



𝑟 → 0(

ğ‘Ž1𝑟+ ⋅ ⋅ ⋅ + ğ‘Žğ‘›ğ‘Ÿ

𝑛 )


= (ğ‘Ž1. . . ğ‘Žğ‘›)1/𝑛 (14)

for allğ‘Ž1, . . . , ğ‘Žğ‘› > 0.

The inequalities (7) can be written asP−1 < P0 < P1.

These inequalities hold for any number of variables and they

follow from the more general fact that P𝑟(ğ‘Ž1, . . . , ğ‘Žğ‘›), for

fixedğ‘Ž1, . . . , ğ‘Žğ‘› > 0, is strictly increasing with𝑟. 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 isclosestto 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.

If𝑑is a distance on𝑆, and if𝐴 = (ğ‘Ž1, . . . , ğ‘Žğ‘›)is a data set

in𝑆, then a𝑑-meanof𝐴is defined to be any element of𝑆at

which the function

𝑓 (𝑥) =∑𝑛


𝑑 (𝑥, ğ‘Žğ‘–) (15)

attains its minimum. It is conceivable that (15) attains its

min-imum at many points, or nowhere at all. However, we shall be

mainly interested in distances𝑑onJfor which (15) attains

its minimum at a unique point𝑥𝐴that, furthermore, has the


min{ğ‘Ž : ğ‘Ž ∈ 𝐴} ≤ 𝑥𝐴≤max{ğ‘Ž : ğ‘Ž ∈ 𝐴} (16)

for every data set𝐴. Such a distance is called a

mean-produc-ingor amean-definingdistance, and the point𝑥𝐴is called the

𝑑-meanof𝐴or themean of𝐴arising from the distance𝑑and

will be denoted by𝜇𝑑(𝐴). A meanMis called adistance mean

if it is of the form𝜇𝑑for some distance𝑑.

Problem Set 1. (1-a) Characterize those distances onJthat are mean-producing.

(1-b) Characterize those pairs of mean producing

distan-ces onJthat produce the same mean.

(1-c) Characterize distance means.

4. Examples of Mean-Producing Distances

If𝑑0is the discrete metric defined onRby

𝑑0(ğ‘Ž, 𝑏) = {10 ifğ‘Ž ̸= 𝑏,

ifğ‘Ž = 𝑏, (17)

then the function𝑓(𝑥) in (15) is nothing but the number

of elements in the given data set𝐴that are different from

𝑥, and therefore every element having maximum frequency

in𝐴minimizes (15) and is hence a𝑑0-mean of𝐴. Thus the


“the”modeof𝐴. Due to the nonuniqueness of the mode, the discrete metric is not a mean-producing distance.

Similarly, the usual metric𝑑 = 𝑑1defined onRby

𝑑1(ğ‘Ž, 𝑏) = |ğ‘Ž − 𝑏| (18)

is not a mean-producing distance. In fact, it is not very

diffi-cult to see that if𝐴 = (ğ‘Ž1, . . . , ğ‘Žğ‘›)is an ordered data set of

even size𝑛 = 2𝑚, then any number in the closed interval

[ğ‘Žğ‘š, ğ‘Žğ‘š+1]minimizes



𝑗=1󵄨󵄨󵄨󵄨󵄨𝑥 − ğ‘Žğ‘—óµ„¨óµ„¨óµ„¨óµ„¨óµ„¨ (19)

and is therefore a𝑑1-mean of𝐴. Similarly, one can show that

if𝐴is of an odd size𝑛 = 2𝑚 − 1, thenğ‘Žğ‘šis the unique𝑑1

-mean of𝐴. Thus the usual metric onRgives rise to what is

referred to in statistics as “the”medianof𝐴.

On the other hand, the distance𝑑2defined onRby

𝑑2(ğ‘Ž, 𝑏) = (ğ‘Ž − 𝑏)2 (20)

is a mean-producing distance, although it is not a metric. In fact, it follows from simple derivative considerations that the function




(𝑥 − ğ‘Žğ‘—)2 (21)

attains its minimum at the unique point

𝑥 = 1 𝑛(




ğ‘Žğ‘—) . (22)

Thus𝑑2is 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-called

midrange, will be encountered below as a very naturallimiting

distance mean.

The distances𝑑1and𝑑2(and in a sense,𝑑0also) are

mem-bers of the family𝑑𝑝of distances defined by

𝑑𝑝(ğ‘Ž, 𝑏) = |ğ‘Ž − 𝑏|𝑝. (23)

It is not difficult to see that if𝑝 > 1, then𝑑𝑝is a

mean-produc-ing distance. In fact, if𝐴 = (ğ‘Ž1, . . . , ğ‘Žğ‘›)is a given data set, and


𝑓 (𝑥) =∑𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑥 − ğ‘Žğ‘—óµ„¨óµ„¨óµ„¨óµ„¨óµ„¨ 𝑝

, (24)


𝑓󸀠󸀠(𝑥) = 𝑝 (𝑝 − 1)∑𝑛

𝑗=1󵄨󵄨󵄨󵄨󵄨𝑥 − ğ‘Žğ‘—óµ„¨óµ„¨óµ„¨óµ„¨óµ„¨ 𝑝−2

≥ 0, (25)

with equality if and only ifğ‘Ž1 = ⋅ ⋅ ⋅ = ğ‘Žğ‘› = 𝑥. Thus𝑓

is convex and cannot attain its minimum at more than one point. That it attains its minimum follows from the continuity

of𝑓(𝑥), the compactness of[ğ‘Ž1, ğ‘Žğ‘›], and the obvious fact that

𝑓(𝑥)is increasing on[ğ‘Žğ‘›, ∞)and is decreasing on(−∞, ğ‘Ž1].

If we denote the mean that𝑑𝑝defines by𝜇𝑝, then𝜇𝑝(𝐴)is the

unique zero of




sign(𝑥 − ğ‘Žğ‘—) 󵄨󵄨󵄨󵄨󵄨𝑥 − ğ‘Žğ‘—óµ„¨óµ„¨óµ„¨óµ„¨óµ„¨ğ‘âˆ’1, (26)

where sign(𝑡) is defined to be 1 if𝑡 is nonnegative and−1


Note that no matter what𝑝 > 1is, the two-dimensional

mean𝜇𝑝arising from𝑑𝑝is the arithmetic mean. Thus when

studying𝜇𝑝, we confine our attention to the case when the

number𝑘of variables is greater than two. For such𝑘, it is

impossible in general to compute𝜇𝑝(𝐴)in closed form.

Problem 2. It would be interesting to investigate

comparabil-ity among{𝜇𝑝: 𝑝 > 1}.

It is highly likely that no two means𝜇𝑝are comparable.

5. Deviation and Sparseness

If 𝑑 is a mean-producing distance on 𝑆, and if 𝜇𝑑 is the

associated mean, then it is natural to define the𝑑-deviation

D𝑑(𝐴)of a data set𝐴 = (ğ‘Ž1, . . . , ğ‘Žğ‘›)by an expression like

D𝑑(𝐴) = 𝜇𝑑{𝑑 (𝜇𝑑(𝐴) , ğ‘Žğ‘–) : 1 ≤ 𝑖 ≤ 𝑛} . (27)

Thus if𝑑is defined by

𝑑 (𝑥, 𝑦) = (𝑥 − 𝑦)2, (28)

then 𝜇𝑑 is nothing but the arithmetic mean or ordinary

average𝜇defined by

𝜇 = 𝜇 (ğ‘Ž1, . . . , ğ‘Žğ‘›) = ğ‘Ž1+ ⋅ ⋅ ⋅ + ğ‘Žğ‘›

𝑛 , (29)

andD𝑑is the (squared) standard deviationğœŽ(2)given by

ğœŽ(2)(ğ‘Ž1, . . . , ğ‘Žğ‘›) = óµ„¨óµ„¨óµ„¨óµ„¨ğ‘Ž1− 𝜇󵄨󵄨󵄨󵄨


+ ⋅ ⋅ ⋅ + óµ„¨óµ„¨óµ„¨óµ„¨ğ‘Žğ‘›âˆ’ 𝜇󵄨󵄨󵄨󵄨2

𝑛 . (30)

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 ğœŽ(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 of𝜇given in (29), accepting the

standard definition of the standard deviation given in (30) as


It is worth mentioning that choosing an exponent other

than 2 in (30) would result in anessentiallydifferent notion

of deviations. More precisely, if one definesğœŽ(𝑘)by


1, . . . , ğ‘Žğ‘›) = óµ„¨óµ„¨óµ„¨óµ„¨ğ‘Ž1− 𝜇󵄨󵄨󵄨󵄨 𝑘

+ ⋅ ⋅ ⋅ + óµ„¨óµ„¨óµ„¨óµ„¨ğ‘Žğ‘›âˆ’ 𝜇󵄨󵄨󵄨󵄨𝑘

𝑛 , (31)

then ğœŽ(𝑘) and ğœŽ(2) would of course be unequal, but more

importantly, they would not bemonotonewith respect to each

other, in the sense that there would exist data sets𝐴and𝐵

withğœŽ(2)(𝐴) > ğœŽ(𝑘)(𝐵)andğœŽ(2)(𝐴) < ğœŽ(𝑘)(𝐵). Thus the choice

of the exponent𝑘in 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. Let 𝑑 be the distance defined by 𝑑(𝑥, 𝑦) =

|𝑥 − 𝑦|𝑘, and let the associated deviationD𝑑defined in (27)

be denoted byD𝑘. IsD𝑘monotone with respect toD2for any

𝑘 ̸= 2, in the sense that

D𝑘(𝐴) >D𝑘(𝐵) 󳨐⇒D2(𝐴) >D2(𝐵)? (32)

We end this section by introducing the notion of

sparse-nessand by observing its relation with deviation. If 𝑑is a

mean-producing distance onJ, and if𝜇𝑑 is the associated

mean, then the 𝑑-sparseness S𝑑(𝐴) of a data set 𝐴 =

(ğ‘Ž1, . . . , ğ‘Žğ‘›)inJcan be defined by

S𝑑(𝐴) = 𝜇𝑑{𝑑 (ğ‘Žğ‘–, ğ‘Žğ‘—) : 1 ≤ 𝑖 < 𝑗 ≤ 𝑛} . (33)

It is interesting that when𝑑is defined by (28), the standard

deviation coincides, up to a constant multiple, with the sparsenss. One wonders whether this pleasant property

char-acterizes this distance𝑑.

Problem Set 4. (4-a) Characterize those mean-producing dis-tances whose associated mean is the arithmetic mean.

(4-b) If𝑑is as defined in (28), and if𝑑󸀠is another

mean-producing distance whose associated mean is the arithmetic

mean, does it follow thatD𝑑󸀠 and D𝑑 are monotone with

respect to each other?

(4-c) Characterize those mean-producing distances𝛿for

which the deviationD𝛿(𝐴)is determined by the sparseness

S𝛿(𝐴)for every data set𝐴, and vice versa.

6. Best Approximation Means

It is quite transparent that the discussion in the previous

sec-tion regarding the distance mean𝜇𝑝,𝑝 > 1, can be written

in terms of best approximation inℓ𝑛𝑝, the vector spaceR𝑛

endowed with the𝑝-norm‖ ⋅ ⋅ ⋅ ‖𝑝defined by

󵄩󵄩󵄩󵄩(ğ‘Ž1, . . . , ğ‘Žğ‘›)󵄩󵄩󵄩󵄩𝑝= ( 𝑛


𝑗=1óµ„¨óµ„¨óµ„¨óµ„¨óµ„¨ğ‘Žğ‘—óµ„¨óµ„¨óµ„¨óµ„¨óµ„¨ 𝑝



. (34)

If we denote byΔ = Δ𝑛 the line in R𝑛 consisting of the

points (𝑥1, . . . , 𝑥𝑛) with 𝑥1 = ⋅ ⋅ ⋅ = 𝑥𝑛, then to say that

ğ‘Ž = 𝜇𝑝(ğ‘Ž1, . . . , ğ‘Žğ‘›)is just another way of saying that the point

(ğ‘Ž, . . . , ğ‘Ž)is a best approximant inΔ𝑛of the point(ğ‘Ž1, . . . , ğ‘Žğ‘›)

with respect to the𝑝-norm given in (34). Here, a point𝑠𝑡in

a subset𝑆of a metric (or distance) space(𝑇, 𝐷)is said to be a

best approximantin𝑆of𝑡 ∈ 𝑇if𝐷(𝑡, 𝑠𝑡) =min{𝐷(𝑡, 𝑠) : 𝑠 ∈

𝑆}. Also, a subset𝑆of(𝑇, 𝐷)is said to beChebyshevif every𝑡

in𝑇has exactly one best approximant in𝑆; see [4, p. 21].

The discussion above motivates the following definition.

Definition 1. LetJbe an interval inRand let𝐷be a distance

onJ𝑛. If the diagonalΔ(J𝑛)ofJ𝑛defined by

Δ (J𝑛) = {(ğ‘Ž1, . . . , ğ‘Žğ‘›) ∈J𝑛: ğ‘Ž1= ⋅ ⋅ ⋅ = ğ‘Žğ‘›} (35)

is Chebyshev (with respect to 𝐷), then the 𝑛-dimensional

mean𝑀𝐷onJdefined by declaring𝑀𝐷(ğ‘Ž1, . . . , ğ‘Žğ‘›) = ğ‘Žif

and only if(ğ‘Ž, . . . , ğ‘Ž)is the best approximant of(ğ‘Ž1, . . . , ğ‘Žğ‘›)in

Δ(J𝑛)is called theChebyshevorbest approximation𝐷-mean

or thebest approximation mean arising from𝐷.

In particular, if one denotes by𝑀𝑝the best approximation

𝑛-dimensional mean onRarising from (the distance onR𝑛

induced by) the norm‖ ⋅ ⋅ ⋅ ‖𝑝, then the discussion above says

that𝑀𝑝exists for all𝑝 > 1and that it is equal to𝜇𝑝defined

inSection 4.

In view of this, one may also defineğ‘€âˆž to be the best

approximation mean arising from the∞-norm ofâ„“âˆžğ‘› , that is,

the norm‖ ⋅ ⋅ ⋅ ‖∞defined onR𝑛by

󵄩󵄩󵄩󵄩(ğ‘Ž1, . . . , ğ‘Žğ‘›)󵄩󵄩󵄩󵄩∞=max{óµ„¨óµ„¨óµ„¨óµ„¨óµ„¨ğ‘Žğ‘—óµ„¨óµ„¨óµ„¨óµ„¨óµ„¨ : 1 ≤ 𝑗 ≤ 𝑛} . (36)

It is not very difficult to see thatğœ‡âˆž(𝐴)is nothing but what

is referred to in statistics as themid-rangeof𝐴. Thus if𝐴 =

(ğ‘Ž1, . . . , ğ‘Žğ‘›)is an ordered data set, then

ğ‘€âˆž(𝐴) = ğ‘Ž1+ ğ‘Ž2 𝑛. (37)

In view of the fact thatğ‘‘âˆžcannot be defined by anything like

(23) andğœ‡âˆžis thus meaningless, natural question arises as to


ğ‘€âˆž(𝐴) = lim

𝑝 → âˆžğœ‡ğ‘(𝐴) (or equivalently=𝑝 → ∞lim 𝑀𝑝(𝐴))


for every 𝐴. An affirmative answer is established in [5,

Theorem 1]. In that theorem, it is also established that


𝑝 → ğ‘žğœ‡ğ‘(𝐴) (or equivalently lim𝑝 → ğ‘žğ‘€ğ‘(𝐴)) = ğ‘€ğ‘ž(𝐴)


for allğ‘žand all𝐴. All of this can be expressed by saying that

𝜇𝑝is continuous in𝑝for𝑝 ∈ (1, ∞]for all𝐴.

We remark that there is no obvious reason why (38)

should immediately follow from the well known fact that


𝑝 → âˆžâ€–ğ´â€–ğ‘= â€–ğ´â€–âˆž (40)


Problem Set 5. Suppose that𝛿𝑝is a sequence of distances on

a set𝑆 that converges to a distance ğ›¿âˆž (in the sense that

lim𝑝 → âˆžğ›¿ğ‘(ğ‘Ž, 𝑏) = ğ›¿âˆž(ğ‘Ž, 𝑏)for allğ‘Ž,𝑏in𝑆). Let𝑇 ⊆ 𝑆.

(5-a) If 𝑇 is Chebyshev with respect to each 𝛿𝑝, is it

necessarily true that𝑇is Chebyshev with respect to


(5-b) If𝑇is Chebyshev with respect to each 𝛿𝑝 and with

respect toğ›¿âˆžand if𝑥𝑝is the best approximant in𝑇of

𝑥with respect to𝛿𝑝andğ‘¥âˆžis the best approximant

in𝑇of𝑥with respect toğ›¿âˆž, does it follow that𝑥𝑝

converges toğ‘¥âˆž?

We end this section by remarking that if 𝑀 = 𝑀𝑑 is

the𝑛-dimensional best approximation mean arising from a

distance𝑑onJ𝑛, then𝑑is significant only up to its values of

the type𝑑(𝑢,V), where𝑢 ∈ Δ(J𝑛)andV∉ Δ(J𝑛). Other values

of𝑑are not significant. This, together with the fact that

every mean is a best approximation mean arising

from a metric, (41)

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 mean M 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 of𝐷in [6] and see what

other nice properties (such as continuity with respect to the

usual metric)𝐷has. This would restore merit to the

construc-tion in [6] and to the proofs therein and provide raison d’ˆetre

for the so-calledgeneralizedmeans introduced there.

7. Towards a Unique Median

As mentioned earlier, the distance𝑑1onRdefined by (23)

does not give rise to a (distance) mean. Equivalently, the 1-norm‖ ⋅ ⋅ ⋅ ‖1 onR𝑛 defined by (34) does not give rise to a (best approximation) mean. These give rise, instead, to the

many-valued function known asthe median. Thus, following

the statistician’s mode of thinking, one may set

𝜇1(𝐴) = 𝑀1(𝐴) = the median interval of𝐴

= the set of all medians of𝐴. (42)

From a mathematician’s point of view, however, this leaves a lot to be desired, to say the least. The feasibility and naturality

of definingğœ‡âˆž as the limit of𝜇𝑝 as𝑝approaches∞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 of𝜇𝑝(𝐴)

(equivalently of𝑀𝑝(𝐴)) as𝑝decreases to 1 exists for every

𝐴 ∈R𝑛and equals one of the medians described in (42). This

limit can certainly be used asthedefinition ofthemedian.

Problem Set 7. Let𝜇𝑝be as defined inSection 4, and let𝜇∗be

the limit of𝜇𝑝as𝑝decreases to 1.

(7-a) Explore how the value of𝜇∗(𝐴)compares with the

common practice of taking the median of𝐴to be the

midpoint of the median interval (defined in (42) for

various values of𝐴.

(7-b) Is𝜇∗continuous onR𝑛? If not, what are its points of


(7-c) Given 𝐴 ∈ R𝑛, is the convergence of 𝜇𝑝(𝐴)(as 𝑝

decreases to 1) to𝜇∗(𝐴)monotone?

The convergence of𝜇𝑝(𝐴)(as𝑝decreases to 1) to𝜇∗(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


It is of course legitimate to question the usefulness of

defining the median to be𝜇∗, but that can be left to

statis-ticians 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 to𝑑𝑝, a sequence𝑑󸀠𝑝of distances onRthat converges

to 𝑑1 such that the limit 𝜇∗∗, as𝑝 decreases to 1, of their

associated distance means𝜇󸀠𝑝is not the same as the limit𝜇∗of

𝜇𝑝. In this case,𝜇∗∗would have as valid a claim as𝜇∗to being

themedian. However, the naturality of𝑑𝑝may help accepting

𝜇∗as a most legitimate median.

Problem Set 8. Suppose that𝛿𝑝and𝛿󸀠𝑝,𝑝 ∈N, are sequences

of distances on a set 𝑆 that converge to the distances ğ›¿âˆž

and ğ›¿âˆžó¸€  , respectively (in the sense that lim𝑝 → âˆžğ›¿ğ‘(ğ‘Ž, 𝑏) =

ğ›¿âˆž(ğ‘Ž, 𝑏)for allğ‘Ž, 𝑏in𝑆, etc.).

(8-a) If each𝛿𝑝, 𝑝 ∈ N, is mean producing with

corre-sponding mean𝑚𝑝, does it follow thatğ›¿âˆžis mean

producing? If so, and if the mean produced byğ›¿âˆžis

ğ‘šâˆž, is it necessarily true that𝑚𝑝converges toğ‘šâˆž?

(8-b) If 𝛿𝑝 and 𝛿󸀠𝑝, 𝑝 ∈ N ∪ {∞}, are mean producing

distances with corresponding means𝑚𝑝and𝑚󸀠𝑝, and

if𝑚𝑝 = 𝑚󸀠𝑝 for all𝑝 ∈ N, does it follow thatğ‘šâˆž =

𝑚󸀠 ∞?

8. Examples of Distance Means

It is clear that the arithmetic mean is the distance mean

arising from the the distance𝑑2given by𝑑2(ğ‘Ž, 𝑏) = (ğ‘Ž − 𝑏)2.

Similarly, the geometric mean on the set of positive numbers

is the distance mean arising from the distance𝑑Ggiven by

𝑑G(ğ‘Ž, 𝑏) = (lnğ‘Ž −ln𝑏)2. (43)

In fact, this should not be amazing since the arithmetic mean

AonRand the geometric meanGon(0, ∞)areequivalent

in the sense that there is a bijection𝑔 : (0, ∞) → R, namely

𝑔(𝑥) = ln𝑥, for whichG(ğ‘Ž, 𝑏) = 𝑔−1A(𝑔(ğ‘Ž), 𝑔(𝑏))for all

ğ‘Ž, 𝑏. Similarly, the harmonic and arithmetic means on(0, ∞)


the harmonic mean is the distance mean arising from the

distance𝑑Hgiven by

𝑑H(ğ‘Ž, 𝑏) = (1ğ‘Žâˆ’1 𝑏)


. (44)

The analogous question pertaining to the logarithmic mean

Ldefined by

L(ğ‘Ž, 𝑏) = ğ‘Ž − 𝑏

lnğ‘Ž −ln𝑏, ğ‘Ž, 𝑏 > 0, (45)

remains open.

Problem 9. Decide whether the meanL(defined in (45)) is a distance mean.

9. Quasi-Arithmetic Means

A𝑘-dimensional meanMonJ is called aquasi-arithmetic

mean if there is a continuous strictly monotone function𝑔

fromJto an intervalIinRsuch that

M(ğ‘Ž1, . . . , ğ‘Žğ‘˜) = 𝑔−1(A(𝑔 (ğ‘Ž1) , . . . , 𝑔 (ğ‘Žğ‘˜))) (46)

for allğ‘Žğ‘—inJ. We have seen that the geometric and harmonic

means are quasi-arithmetic and concluded that they are

distance means. To see that Lis not quasi-arithmetic, we

observe that the (two-dimensional) arithmetic mean, and

hence any quasi-arithmetic mean M, satisfies the elegant

functional equation

M(M(M(ğ‘Ž, 𝑏) , 𝑏) ,M(M(ğ‘Ž, 𝑏) , ğ‘Ž)) =M(ğ‘Ž, 𝑏) (47)

for allğ‘Ž, 𝑏 > 0. However, a quick experimentation with a

random pair(ğ‘Ž, 𝑏)shows that (47) is not satisfied byL.

This shows thatLis not quasi-arithmetic, but does not

tell us whether Lis a distance mean, and hence does not

answer Problem9.

The functional equation (47) is a weaker form of the

functional equation

M(M(ğ‘Ž, 𝑏) ,M(𝑐, 𝑑)) =M(M(ğ‘Ž, 𝑐) ,M(𝑏, 𝑑)) (48)

for all ğ‘Ž, 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 meanMthat satisfies the

func-tional 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 function 𝐸 = 𝐸(𝑥, 𝑡) onR2 is called a

deviationif𝐸(𝑥, 𝑥) = 0for all𝑥 and if𝐸(𝑥, 𝑡)is a strictly

decreasing continuous function of 𝑡 for every 𝑥. If 𝐸 is a

deviation, and if𝑥1, . . . , 𝑥𝑛 are given, then the𝐸-deviation

meanof𝑥1, . . . , 𝑥𝑛is defined to be the unique zero of

𝐸 (𝑥1, 𝑡) + ⋅ ⋅ ⋅ + 𝐸 (𝑥𝑛, 𝑡) . (49)

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.

If𝐸is a deviation, then (following [11]), one may define


𝑑𝐸(𝑥, 𝑡) = ∫ 𝑡

𝑥𝐸 (𝑥, 𝑠) 𝑑𝑠. (50)

Then𝑑𝐸(𝑥, 𝑡) ≥ 0and𝑑𝐸(𝑥, 𝑡)is a strictly convex function in𝑡

for every𝑥. The𝐸-deviation mean of𝑥1, . . . , 𝑥𝑛is nothing but

the unique value of𝑡at which𝑑𝐸(𝑥1, 𝑡)+⋅ ⋅ ⋅+𝑑𝐸(𝑥𝑛, 𝑡)attains

its minimum. Thus if𝑑𝐸happens to be symmetric, then𝑑𝐸

would be a distance and the𝐸-deviation mean would be the

distance mean arising from the distance𝑑𝐸.

11. Other Ways of Generating New Means

If𝑓and𝑔are differentiable on an open intervalJ, and ifğ‘Ž <

𝑏are points inJsuch that𝑓(𝑏) ̸= 𝑓(ğ‘Ž), then there exists, by

Cauchy’s mean value theorem, a point𝑐in(ğ‘Ž, 𝑏), such that


𝑔󸀠(𝑐) =

𝑔 (𝑏) − 𝑔 (ğ‘Ž)

𝑓 (𝑏) − 𝑓 (ğ‘Ž). (51)

If𝑓and𝑔are such that𝑐is unique for everyğ‘Ž, 𝑏, then we call

𝑐theCauchymean ofğ‘Žand𝑏corresponding to the functions

𝑓and𝑔, and we denote it byC𝑓,𝑔(ğ‘Ž, 𝑏).

Another natural way of defining means is to take a

continuous function𝐹that is strictly monotone onJ, and to

define the mean ofğ‘Ž, 𝑏 ∈J,ğ‘Ž ̸= 𝑏, to be the unique point𝑐in

(ğ‘Ž, 𝑏)such that

𝐹 (𝑐) = 𝑏 − ğ‘Ž1 ∫𝑏

ğ‘Ž 𝐹 (𝑥) 𝑑𝑥. (52)

We call𝑐themean value(mean) ofğ‘Žand𝑏corresponding to

𝐹, and we denote it byV(ğ‘Ž, 𝑏).

Clearly, if𝐻is an antiderivative of𝐹, then (53) can be

written as

𝐻󸀠(𝑐) = 𝐻 (𝑏) − 𝐻 (ğ‘Ž)

𝑏 − ğ‘Ž . (53)

ThusV𝐹(ğ‘Ž, 𝑏) =C𝐻,𝐸(ğ‘Ž, 𝑏), where𝐸is 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 function

𝑓onJ𝑛 may be called aChisini function if and only if the



has a unique solution𝑥 = ğ‘Ž ∈ [ğ‘Ž1, ğ‘Žğ‘›]for every ordered data set(ğ‘Ž1, . . . , ğ‘Žğ‘›)inJ. This unique solution is called theChisini

meanassociated to𝑓. In Chisini’s own words,𝑥is said to be

the mean of 𝑛numbers𝑥1, . . . , 𝑥𝑛 with respect to a problem, in which a function of them𝑓(𝑥1, . . . , 𝑥𝑛)is of interest, if the function assumes the same value when all theğ‘¥â„Žare 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, devia-tion, 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 → Jto be a mean is to satisfy the internality


min{ğ‘Ž1, . . . , ğ‘Žğ‘˜} ≤M(ğ‘Ž1, . . . , ğ‘Žğ‘˜) ≤max{ğ‘Ž1, . . . , ğ‘Žğ‘˜} (55)

for allğ‘Žğ‘— ∈J. However, one may ask whether it is sufficient,

for certain types of functionsM, to verify (55) for a finite,

preferably small, number of well-chosen𝑛-tuples. This

ques-tion 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)formin𝑛variables, we shall always mean a

homoge-neous polynomial𝐹 = 𝐹(𝑥1, . . . , 𝑥𝑛)in the indeterminates

𝑥1, . . . , 𝑥𝑛 having coefficients inR. When the degree 𝑡of a

form𝐹is to be emphasized, we call 𝐹a𝑡-form. Forms of

degrees 1, 2, 3, 4, and 5 are referred to aslinear, quadratic,

cubic,quartic, andquinticforms, respectively.

The set of all𝑡-forms in𝑛variables is a vector space (over

R) that we shall denote by F(𝑛)

𝑡 . It may turn out to be an

interesting exercise to prove that the set

{ { {





𝑗 : 𝑑



𝑗𝑒𝑗= 𝑑}} }


is a basis, where𝑁𝑗is the Newton polynomial defined by

𝑁𝑗 =∑𝑛


𝑥𝑗𝑘. (57)

The statement above is quite easy to prove in the special case

𝑑 ≤ 3, and this is the case we are interested in in this paper.

We also discard the trivial case𝑛 = 1and assume always that

𝑛 ≥ 2.

Linear forms can be written as ğ‘Žğ‘1, and they are not

worth much investigation. Quadratic forms can be written as

𝑄 = ğ‘Žğ‘12+ 𝑏𝑁2= ğ‘Ž(∑𝑛




+ 𝑏 (∑𝑛


𝑥2𝑘) . (58)

Cubic and quartic forms can be written, respectively, as

ğ‘Žğ‘13+ 𝑏𝑁1𝑁2+ 𝑐𝑁3,


1 + 𝑏𝑁12𝑁2+ 𝑐𝑁1𝑁3+ 𝑑𝑁22.


A form 𝐹 = 𝐹(𝑥1, . . . , 𝑥𝑛) is said to be copositive if

𝑓(ğ‘Ž1, . . . , ğ‘Žğ‘›) ≥ 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 number𝑛 ≥ 2of variables. Letv𝑚(𝑛),1 ≤ 𝑚 ≤ 𝑛, be the 𝑛-tuple whose first𝑚coordinates are 1’s and whose remaining coordinates are 0󸀠s.

(i)If𝐹is quadratic, then𝐹is copositive if and only if𝐹 ≥ 0 at the two test𝑛-tuples

k(𝑛)1 = (1, 0, . . . , 0) , k𝑛(𝑛)= (1, 1, . . . , 1) . (60)

(ii)If𝐹is cubic, then𝐹is copositive if and only if𝐹 ≥ 0at the𝑛test𝑛-tuples

k(𝑛)𝑚 = (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1, . . . , 1,𝑚 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, . . . , 0) , 1 ≤ 𝑚 ≤ 𝑛.𝑛−𝑚 (61)

Part (i) is a restatement of Theorem 1(a) in [16].

Theo-rem 1(b) there is related and can be restated as

𝐹 (ğ‘Ž1, . . . , ğ‘Žğ‘›) ≥ 0, âˆ€ğ‘Žğ‘–âˆˆR, ⇐⇒ 𝐹 ≥ 0at the3 𝑛-tuples

(1, 0, . . . , 0) , (1, 1, . . . , 1) , (1, −1, 0, . . . , 0) .


Part (ii) was proved in [17] for𝑛 ≤ 3and in [18] for all𝑛. Two

very short and elementary inductive proofs are given in [19].

It is worth mentioning that the 𝑛test 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 form𝑓

in𝑛variables is copositive that consists in testing𝑓at𝑛-tuples

of the type

(âžâžâžâžâžâžâžâžâžâžâžâžâžğ‘Ž, . . . , ğ‘Ž,𝑚 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1, . . . , 1,𝑟 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, . . . , 0) ,𝑛−𝑚−𝑟

0 ≤ 𝑚, 𝑟 ≤ 𝑛, 𝑚 + 𝑟 ≤ 𝑛


is established in [21]. It is also proved there that if𝑛 = 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 real𝑡-forms in𝑛

variables, and let𝑁𝑗,1 ≤ 𝑗 ≤ 𝑑, be the Newton polynomials

defined in (57). Means of the type

M= (𝐹𝑟




where𝐹𝑗 is a symmetric form of degree𝑗, are clearly sym-metric and 1-homogeneous, and they abound in the literature.

These include the family of Gini means𝐺𝑟,𝑠defined in (8)

(and hence the Lehmer and H¨older means). They also include the elementary symmetric polynomial and elementary

sym-metric polynomial ratio means defined earlier in (10).

In view ofTheorem 2of the previous section, it is

tempt-ing to ask whether the internality of a functionMof the type

described in (64) can be established by testing it at a finite

set of test𝑛-tuples. Positive answers for some special cases of

(64), and for other related types, are given in the following


Theorem 3. Let𝐿,𝑄, and𝐶be real symmetric forms of degrees

1, 2, and 3, respectively, in any number𝑛 ≥ 2of nonnegative variables. Letv(𝑛)𝑘 ,1 ≤ 𝑘 ≤ 𝑛, be as defined inTheorem 2.

(i)√𝑄is internal if and only if it is internal at the two test 𝑛-tuples:k(𝑛)𝑛 = (1, 1, . . . , 1)andV(𝑛)𝑛−1= (1, 1, . . . , 1, 0).

(ii)𝑄/𝐿is internal if and only if it is internal at the two test 𝑛-tuples:k(𝑛)

𝑛 = (1, 1, . . . , 1)andV(𝑛)1 = (1, 0, . . . , 0).

(iii)If𝑛 ≤ 4, then√𝐶3 is internal if and only if it is internal

at the𝑛test𝑛-tuples


𝑚 = (



1, . . . , 1,⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, . . . , 0) , 1 ≤ 𝑚 ≤ 𝑛.𝑛−𝑚 (65)

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 ≤ 4only. The problem for𝑛 ≥ 5, together with other open problems, is listed in the next problem set.

Problem Set 13. Let𝐿,𝑄, and𝐶be real symmetric cubic forms

of degrees 1, 2, and 3, respectively, in𝑛non-negative variables.

(13-a) Prove or disprove that√𝐶3 is internal if and only if it

is internal at the𝑛test𝑛-tuples

k(𝑛)𝑚 = (⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞1, . . . , 1,𝑚 ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞0, . . . , 0) , 1 ≤ 𝑚 ≤ 𝑛.𝑛−𝑚 (66)

(13-b) Find, or prove the nonexistence of, a finite set 𝑇of

test𝑛-tuples such that the internality of𝐶/𝑄at the𝑛

-tuples in𝑇gurantees its internality at all nonnegative


(13-c) Find, or prove the nonexistence of, a finite set 𝑇of

test𝑛-tuples such that the internality of𝐿 ± √𝑄at

the𝑛-tuples in𝑇guarantees its internality at all


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 inTheorem 2there that when𝑄has the special form

ğ‘Žâˆ1≤𝑖<𝑗≤𝑛(𝑥𝑖 − 𝑥𝑗)2, then 𝐿 ± √𝑄 is internal if and only

if it is internal at the two test𝑛-tuplesk𝑛(𝑛) = (1, 1, . . . , 1)

andk𝑛−1(𝑛) = (1, 1, . . . , 1, 0). In the general case, sufficient and

necessary conditions for internality of𝐿 ± √𝑄, in terms of

the coefficients of𝐿 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

A(2)(ğ‘Ž1, ğ‘Ž2) = ğ‘Ž1+ ğ‘Ž2

2 (67)

can be extended to any dimension𝑘by setting

A(𝑘)(ğ‘Ž1, . . . , ğ‘Žğ‘˜) = ğ‘Ž1+ ⋅ ⋅ ⋅ + ğ‘Žğ‘˜

𝑘 . (68)

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 sequence𝐴(𝑘)of meansharmonious

orconcordant. This does not seem to be an acceptable defini-tion of the nodefini-tion of concordance.

In a private communication several years ago, Professor Zsolt P´ales told me that Kolmogorov suggested calling a

sequenceM(𝑘)of means onJ, whereM(𝑘)is𝑘-dimensional,

concordant if for every𝑚and𝑛and everyğ‘Žğ‘–,𝑏𝑖inJ, we have

M(𝑛+𝑚)(ğ‘Ž1, . . . , ğ‘Žğ‘›, 𝑏1, . . . , 𝑏𝑚)

=M(2)(M(𝑛)(ğ‘Ž1, . . . , ğ‘Žğ‘›) ,M𝑚(𝑏1, . . . , 𝑏𝑚)) . (69)

He also told me that such a definition is too restrictive and seems to confirm concordance in the case of the quasi-arith-metic 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 meanLdefined 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

function𝑑on any set𝑋, one may define theopen ball𝐵(ğ‘Ž, 𝑟)

in the usual manner, and then one may declare a subset𝐴 ⊆

𝑋openif it contains, for everyğ‘Ž ∈ 𝐴, an open ball𝐵(ğ‘Ž, 𝑟)with

𝑟 > 0. If𝑑has the triangle inequality, then one can proceed in the usual manner to create a topology. However, for a general

distance𝑑, this need not be the case, and distances that give

rise to a coherent topology in the usual manner are called

semimetricsand they are investigated and characterized in

[24–29]. Clearly, these are the distances𝑑for which the family

{𝐵(ğ‘Ž, 𝑟) : 𝑟 > 0}of open balls centered atğ‘Ž ∈ 𝑆forms a local


15. Centers and Center-Producing Distances

A distance 𝑑 may be defined on any set 𝑆 whatsoever. In

particular, if𝑑is a distance onR2and if the function𝑓(𝑋)

defined by

𝑓 (𝑋) =∑𝑛

𝑖=1𝑑 (𝑋, 𝐴𝑖) (70)

attains its minimum at a unique point𝑋0 that lies in the

convex hull of{𝐴1, . . . , 𝐴𝑛}for every choice of𝐴1, . . . , 𝐴𝑛in

R2, then𝑑will be calleda center-producing distance.

The Euclidean metric 𝑑1 on R2 produces the

Fermat-Torricellicenter. This is defined to be the point whose distan-ces 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 by𝑑𝑝for other values of𝑝.

Problem 15. Let𝑑𝑝,𝑝 > 1, be the distance defined onR2by

𝑑𝑝(𝐴, 𝐵) = ‖𝐴 − 𝐵‖𝑝, and let𝐴𝐵𝐶be a triangle. Let𝑍𝑝 =

𝑍𝑝(𝐴, 𝐵, 𝐶)be the point that minimizes

𝑑𝑝(𝑍, 𝐴) + 𝑑𝑝(𝑍, 𝐵) + 𝑑𝑝(𝑍, 𝐶)

= ‖𝑍 − 𝐴‖𝑝+ ‖𝑍 − 𝐵‖𝑝+ ‖𝑍 − 𝐶‖𝑝. (71)

Investigate how𝑍𝑝,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



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