Multi-argument distances

Fuzzy Sets and Systems 167 (2011) 92–100www.elsevier.com/locate/fss

Multi-argument distances�

J. Martín∗, G. MayorUniversity of the Balearic Islands, Cra. Valldemossa km 7.5, 07112 Palma, Spain

Available online 5 November 2010

Abstract

In this paper, we propose a formal definition of a multi-argument function distance. The conventional definition of a distance isextended to apply to collections of more than two elements. Some general properties of multidistances are investigated and significantexamples are exhibited. Finally, we introduce the concept of ball centered at a list in order to extend the one of the usual ball in ametric space.© 2010 Elsevier B.V. All rights reserved.

Keywords: Distance; Multidistance; Fermat point; Sum-based multidistance; OWA operator

1. Introduction

The conventional definition of distance over a space specifies properties that must be obeyed by any measure of“how separated” two points in that space are. Given a set X, a distance d on X is a function that distinguish betweenevery two different points x and y of X by assigning to the ordered pair (x, y), in a symmetric manner, a single positivereal number, d(x, y), in such a way that, given any point z of X, d(x, y) does not exceed the sum of d(x, z) and d(z, y),and we set d(x, y) = 0 if and only if x = y. These axioms arise when one examines the fundamental properties of thepoint-to-point-along-a straight-line-segment prototype with a view to developing a theory around those properties, atheory that will then be applicable in many situations, some of them very different from that of the prototype.

However one often wants to measure “how separated” the members of a collection of more than two elements are.The conventional way to do this is to combine the pairwise distance values for all pairs of elements in the collection,into an aggregate measure. Thus, given an Euclidean triangle (A, B, C) we can combine the distances AB, AC, BCusing, for instance, a three-dimensional OWA operator, F(x1, x2, x3) = w1 y1 +w2 y2 +w3 y3. Then, we measure “howseparated” (A, B, C) are by means of the formula D(A, B, C) = F(AB, AC, BC). It is clear that we have to choosethe weighting vector (w1, w2, w3) such that the multi-argument distance function D satisfies a group of axioms thatextend to some degree those for ordinary distance functions. For pairwise distances and related distance matrix see forexample [1].

� The authors acknowledge the support of the Govern Balear Grant PCTIB2005GC1-07 and the Spanish DGI Grants MTM2006-08322 andMTM2009-10962. They also wish to thank the referees for their valuable comments and suggestions.

∗ Corresponding author.E-mail address: [email protected] (J. Martín).

0165-0114/$ - see front matter © 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2010.10.018

http://dx.doi.org/10.1016/j.fss.2010.10.018

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J. Martín, G. Mayor / Fuzzy Sets and Systems 167 (2011) 92 –100 93

Of course we can consider many other procedures to measure how separated the vertices (A, B, C) are: In Euclideangeometry the Fermat point of a triangle (A, B, C), also called Torricelli point, is the point F for which the sum of thedistances from F to the vertices is as small as possible; i.e. the Fermat point is the point F such that F A + F B + FCis minimized. Then we can define D(A, B, C) = F A + F B + FC .

In this paper the formal definition of a distance function is extended to apply to collections of more than two elements.The measure presented here applies to n-dimensional ordered lists of elements, and it can be directly incorporated intomany domains where ad hoc combinations of pairwise metrics are currently used. The main advantage of our proposalis that it allows the general treatment of the problem of measuring the distance for more than two points by means ofan axiomatic procedure. This new measure also permits in a natural way to measure how separated from each othertwo collections are.

In the next section the definitions of multidistance and strong multidistance are introduced, and some elementaryproperties of multi-argument distances are presented. Sections 3 and 4 are devoted to present significant examples ofmultidistances: Fermat and averaged multidistances. In Section 5 we introduce the fundamental concept of ball centeredat a list in order to deal with topological properties in a multidistance space.

In 2004, Wolpert [9] presented the definition of a multi-argument metric (multimetric, for short) in a rather differentmanner. The measure introduced by Wolpert applies even to collections with “fractional” numbers of elements, howeverhis axiomatic, an extension of the usual conditions defining a metric, is stronger than the one we present here. To bemore precise, Wolpert’s multimetrics can be viewed as similar to our strong multidistances.

In [7,2,4] terms like n-distances and multi-metrics are introduced in certain contexts, but with a very different meaningwith respect to those defined by Wolpert and ourselves. Thus, in [4] a multi-metric space is simply a union X = ⋃

i Xi

where each Xi is a metric space with metric di , i = 1, . . . , m. In [7] given a vector of metric functions (d1, . . . , dm)and one of the weights (w1, . . . , wm), wi ∈ [0, 1], a (linear) multi-metric is defined as d = w1d1 + · · · + wmdm .

2. Multidistances

Definition 1. A function D :⋃

n≥1 Xn → [0, ∞] is a multidistance on a non-empty set X when the followingproperties hold, for all n and x1, . . . , xn, y ∈ X:

(m1) D(x1, . . . , xn) = 0 if and only if xi = x j for all i, j = 1, . . . n.(m2) D(x1, . . . , xn) = D(x�(1), . . . , x�(n)) for any permutation � of 1, . . . , n,(m3) D(x1, . . . , xn) ≤ D(x1, y) + · · · + D(xn, y), We say that D is a strong multidistance if it fulfills (m1), (m2) and:

(m3′) D(x1, . . . xk) ≤ D(x1, y) + · · · + D(xk, y) for all x1, . . . , xk, y ∈ ⋃n≥1 Xn .

Expressions like D(x, y) in (m3′), that is, the function D applied to two lists x = (x1, . . . , xn) ∈ Xn andy = (y1, . . . , ym) ∈ Xm , mean the multidistance of the joined list:

D(x, y) = D(x1, . . . , xn, y1, . . . , ym). (1)

Remark 1.

(1) If D is a multidistance on X, then the restriction of D to X2, D|X2 , is an ordinary distance on X.(2) Any ordinary distance d on X can be extended in order to obtain a multidistance. For example, we can define a

function DM in this way:

DM (x1, . . . , xn) = max{d(xi , x j ); i < j}. (2)

Then, DM is a strong multidistance on X such that D|X2 = d . It will be called maximum multidistance.

Example 1. Based on the drastic distance, defined by

d(x, y) ={

1 if x � y,

0 if x = y,(3)

94 J. Martín, G. Mayor / Fuzzy Sets and Systems 167 (2011) 92 –100

several multidistances can be defined extending it. For instance,

• D1(x1, . . . , xn) ={

0 if xi = x j ∀i, j,

1 otherwise,• D2(x1, . . . , xn) = |{x1, . . . , xn}| − 1.

Both of them are strong.

Proposition 1. Let D and D′ be multidistances on X.

(1) D + D′ is a multidistance on X.(2) If k > 0, then kD is a multidistance on X.(3) D/(1 + D) and min{1, D} are also multidistances on X, with values in [0, 1].

Remark 2. If D is a multidistance on X, then the function D̃ :⋃

n≥1 Xn × ⋃n≥1 Xn → [0, +∞) defined by

D̃(x, y) = D(x1, . . . , xn, y1, . . . , ym) (4)

for all x = (x1, . . . , xn) ∈ Xn and y = (y1, . . . , ym) ∈ Xm is not a distance on the set⋃

n≥1 Xn .However, pseudodistances (distance functions allowing d(x, y) = 0 for distinct values x, y) on this set can be

constructed by means of D and an ordinary distance � on R in this way:

D�(x, y) = �(D(x), D(y)). (5)

3. Fermat multidistances

Let (X, d) be a metric space and (x1, . . . , xn) ∈ Xn .

Proposition 2. The function f : X → [0, +∞) defined by f (x) = ∑ni=1 d(xi , x) is continuous and it reaches a

minimum.

Definition 2. The Fermat set Fx of a list x = (x1, . . . , xn) ∈ Xn is the set of points where the function f given inProposition 2 reaches the minimum value:

Fx ={

x ∈ X:n∑

i=1

d(xi , x) ≤n∑

i=1

d(xi , x ′), ∀x ′ ∈ X

}. (6)

Example 2. In the metric space (Rm, d), where d is the Manhattan distance:

d((a1, . . . , am), (b1, . . . , bm)) =m∑

i=1

|ai − bi |, (7)

the set F associated to the points xi = (ai1, . . . , aim), i = 1, . . . , n is

F(x1,. . .,xn )

=m∏

i=1

[a((n+1)/2�)i , a((n+2)/2�)i ], (8)

where the parentheses in the subindexes mean that a(1)i , . . . , a(m)i is a non-decreasing rearrangement of the componentsi of the points, x� is the floor function and

∏is the usual Cartesian product of sets.

If m is odd, the extreme points of the intervals coincide and the set F(x1,. . .,xn )

reduces to a point, whose coordinates

are the median of the respective coordinates of the points x1, . . . , xn : see Fig. 1.


FF

Fig. 1. Fermat sets F associated to two families of points in R2.

Proposition 3. Let DF :⋃

n≥1 Xn → [0, ∞) be the function defined by

DF (x1, . . . , xn) = minx∈X

{n∑

i=1

d(xi , x)

}. (9)

(1) DF is a multidistance onX.(2) DF is maximum in the sense that any other multidistance D such that D|X2 = DF |X2 takes values not greater

than DF , i.e. D(x) ≤ DF (x), ∀x ∈ Xn .

Proof. Condition (m1): if x1 = · · · = xn , then∑n

i=1 d(xi , x1) = 0 and so DF (x1, . . . , xn) = 0. Conversely, let ussuppose that DF (x1, . . . , xn) = 0 and that there exist two different elements, xr , xs , such that d(xr , xs) > 0; hence, forall x ∈ X, we have a contradiction:∑

d(xi , x) ≥ d(xr , x) + d(xs, x) ≥ d(xr , xs) > 0.

Properties (m2) and (m3), and also (2), follow immediately from (9). �

This function DF will be called the Fermat multidistance associated to an ordinary distance d.

Example 3. Let us consider the drastic distance (3). We denote by x ′1, . . . , x ′

m the elements of the list (x1, . . . , xn) ∈ Xn

without repetitions, arranged in such a way that the respective number of repetitions n1, . . . , nm , verifies n1 ≥ · · · ≥ nm .The elements of the Fermat set associated to (x1, . . . , xn) ∈ Xn are the most repeated elements of the list:

F(x1,. . .,xn )

= {x ′1, . . . , x ′

k : n1 = · · · = nk > nk+1}. (10)

If all of the elements of the list are different, then F(x1,. . .,xn )

= {x1, . . . , xn}.Then the resulting Fermat multidistance is

DF (x1, . . . , xn) = n − n1. (11)

The distance DF is not strong in general, as the following example shows.

Example 4. In R2, with the Euclidean distance:

d((a1, a2), (b1, b2)) =√

(a1 − b1)2 + (a2 − b2)2, (12)

the set F associated to two points A, B is the segment AB. The Fermat set of three points is the Fermat point (see, forexample, [3]), and the Fermat–Weber point, or geometric median, for more than three points [8].

Let us see that the associated DF is not strong. Consider the points A = (1, 0), B = (0, 1), C(−1, 0), D = (0, −1)and E = (0, 0) in R2. The Fermat set of the square (A, B, C, D) is F(A,B,C,D) = {E}, and so DF (A, B, C, D) = 4.


On the other hand, the Fermat points of (A, B, E) and (C, D, E) are ((3 − √3)/6, (3 − √

3)/6) and (−(3 − √3)/6,

−(3 − √3)/6), respectively, and

DF (A, B, E) = DF (C, D, E) < 2.

That is,

DF (A, B, C, D) > DF (A, B, E) + DF (C, D, E),

and DF is not a strong multidistance.

Example 5. The multidistance DF associated to the Manhattan distance of Example 2 is defined in this way:

DF (x1, . . . , xn) =m∑

i=1

n/2�∑j=1

j(a(n− j+1)i − a( j)i ). (13)

4. Sum-based multidistances

Let (X, d) be a metric space. Let us consider the function D :⋃

n≥1 Xn → [0, ∞] defined by{D(x1) = 0,

D(x1, . . . , xn) = ∑i< j d(xi , x j ), n ≥ 2,

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that is, the sum of the pairwise distance values for all pairs of elements of the list. It is not a multidistance: it fulfills(m1) and (m2) but not (m3). For example, let (A, B, C) be an equilateral triangle of side length 1 in the Euclideanplane. Then, condition (m3) is not fulfilled for y = A:

D(A, B, C) = 3�D(A, A) + D(B, A) + D(C, A) = 2.

In order to obtain multidistances based on this sum, we have to multiply it by a factor � which depends on the lengthof the list: � = �(n), n ≥ 1.

Proposition 4. The function D� :⋃

n≥1 Xn → [0, ∞] defined by{D�(x1) = 0,

D�(x1, . . . , xn) = �(n)∑

i< j d(xi , x j ), n ≥ 2(15)

is a multidistance if and only if:

(i) �(2) = 1,(ii) 0 < �(n) ≤ 1/(n − 1) for any n > 2.

Proof. The conditions are necessary: (i) because D�|X2 = d and the first inequality of (ii), 0 < �(n), because D� isnon-negative and fulfills (m1).

Let us see that the second inequality of (ii), �(n) ≤ 1/(n − 1), also is: if not, if there exists n0 such that �(n0) >

1/(n0 − 1), we can consider the drastic distance, two different points x, y ∈ X and (

n0−1︷︸︸︷x, . . . , x, y) ∈ Xn0 :

D�(

n0−1︷︸︸︷x, . . . , x, y) = �(n0) · (n0 − 1) > 1.

But the sum of distances from the components of the list (

n0−1︷︸︸︷x, . . . , x, y) to x is 1, and (m3) does not hold.


Conditions (i) and (ii) are also sufficient: (i) and the first inequality in (ii) ensure D� ≥ 0 and (m1); condition (m2)follows from the definition of D�, and finally, condition (m3) follows from �(n) ≤ 1/(n − 1): for any y ∈ X,

D�(x1, . . . , xn) = �(n)∑i< j

d(xi , x j )

≤ �(n)∑i< j

(d(xi , y) + d(x j , y))

= �(n)(n − 1)∑

i

d(xi , y)

≤∑

i

d(xi , y). �

Some of these multidistances are like averaged sums of the all ordinary distances between all pairs of points. In fact,if �(n) = 1/( n

2 ) ≤ 1/(n − 1), D� is the arithmetic mean of these distances.

4.1. OWA-based multidistances

Let (X; d) be a metric space and W = (Wn)n≥3 a collection of OWA operators Wn with weighting vector(wn

1 , . . . , wnn ). We consider the multi-argument function D :

⋃n≥1 Xn → [0, ∞) defined as follows:⎧⎪⎨⎪⎩

D(x1) = 0,

D(x1, x2) = d(x1, x2),

D(x1, . . . , xn) = Wn(a1, . . . , an) whenever n ≥ 3,

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where ai = D(x1, . . . , x̂i , . . . , xn), i = 1, . . . , n, and (x1, . . . , x̂i , . . . , xn) means the (n − 1)-ordered list obtained fromthe n-list (x1, . . . , xi , . . . , xn) once removed the element placed in the i-position.

To be precise, we will write DW to indicate that this function comes from the collection W .We will write Wn = (wn

1 , . . . , wnn ) for the sake of brevity.

Example 6.

(i) The maximum multidistance DM given by (2) is nothing else that DW with Wn = (0, . . . , 0, 1), for all n ≥ 3.(ii) We can also introduce the median multidistance Dmed , where

Wn ={

(0, . . . , 0, 1, 0, . . . , 0) if n is odd,

(0, . . . , 0, 12 , 1

2 , 0, . . . , 0) if n is even.(17)

(ii) Of course we can consider many other collections (Wn)n≥3; thus we can combine two different classes of OWAoperators:

Wn ={

(0, . . . , 0, 1, 0, . . . , 0) if n is odd,

( 12 , 0, . . . , 0, 1

2 ) if n is even.(18)

Proposition 5. The multi-argument function DW , where W = (Wn)n≥3 is a collection of OWA operatorsWn = (wn

1 , . . . , wnn ) is a multidistance if and only if wn

1 < 1, for all n ≥ 3.

Proof. Let us prove first the “only if part”. Suppose Wn = (1, 0, . . . , 0) for some n ≥ 3, then DW (x1, . . . , xn) =Wn(a1, . . . , an) = min{a1, . . . , an} = 0 does not imply x1 = · · · = xn . Thus we see that DW does not satisfy condition(m1) in Definition 1 and consequently it is not a multidistance.


Reciprocally, let us suppose now Wn = (wn1 , . . . , wn

n ) with wn1 < 1 for all n ≥ 3. We will demonstrate that DW is a

multidistance:Condition (m1), that is, DW (x) = 0 ⇔ xi = x j for all i, j = 1, . . . , n, holds for n=1 and 2. Let us suppose that (m1)

is true for an n ≥ 3 and let us prove it for n + 1. If xi = x j for all i, j = 1, . . . , n + 1, n ≥ 3, then

DW (x1, . . . , xn+1) = Wn+1(a1, . . . , an+1),

where ai = DW (x1, . . . , x̂i , . . . , xn+1), i = 1, . . . , n+1. Therefore ai = 0 for all i = 1, . . . , n+1 and DW (x1, . . . , xn+1)= Wn+1(0, . . . , 0) = 0.

Conversely, if DW (x1, . . . , xn+1) = 0 then

Wn+1(a1, . . . , an+1) = wn+11 a(1) + · · · + wn+1

n+1a(n+1) = 0

(a(1) ≤ · · · ≤ a(n+1) is an increasing reordering of a1, . . . , an+1) thus wn+1i a(i) = 0 for all i = 1, . . . , n + 1,

therefore a(r ) = 0 for some r, 2 ≤ r ≤ n + 1 (recall wn+11 < 1). Let us suppose that a(r ) = as , then we can write

a(r ) = as = DW (x1, . . . , x̂s, . . . , xn+1), therefore DW (x1, . . . , x̂s, . . . , xn+1) = 0, which implies, by the inductionhypothesis, xi = x j for all i, j = 1, . . . , n +1, i, j � s. On the other hand, we know that a(1) ≤ · · · ≤ a(r−1) ≤ a(r ) = 0which finally implies xi = x j for all i, j = 1, . . . , n + 1.

Condition (m2) is DW (x�(1), . . . , x�(n)) = DW (x1, . . . , xn) for all permutation � of 1, 2, . . . , n. Of course this is truefor n = 1 and 2. Let us consider that it is also true for an n ≥ 3 and let us prove it for n + 1:

DW (x�(1), . . . , x�(n+1)) = Wn+1(b1, . . . , bn+1),

where bi = DW (x�(1), . . . , x̂�(i), . . . , x�(n+1)), i = 1, . . . , n + 1.On the other side,

DW (x1, . . . , xn+1) = Wn+1(a1, . . . , an+1),

where ai = DW (x1, . . . , x̂i , . . . , xn+1), i = 1, . . . , n + 1. From the induction hypothesis, we can write bi = a�(i),

i = 1, . . . , n + 1. Finally, from the symmetry of the OWA operators Wn , we have

DW (x�(1), . . . , x�(n+1)) = Wn+1(b1, . . . , bn+1)

= Wn+1(a�(1), . . . , a�(n+1))

= Wn+1(a1, . . . , an+1)

= DW (x1, . . . , xn+1).

And condition (m3), DW (x) ≤ ∑ni=1 DW (xi , y) for all y ∈ X , again by induction. It is true for n = 1 and 2. Let us

consider that (m3) is true for an n ≥ 3 and let us prove it for n + 1.

DW (x1, . . . , xn+1) = Wn+1(a1, . . . , an+1) ≤ max{a1, . . . , an+1}.Let us suppose max{a1, . . . , an+1} = a1, for short. Then we can write, for all y ∈ X ,

DW (x1, . . . , xn+1) ≤ a1 = DW (x2, . . . , xn+1) ≤n∑

i=2

DW (xi , y) ≤n∑

i=1

DW (xi , y). �

5. Balls centered at a list

Given a nonempty set X, a multidistance D defined on it and a list x = (x1, . . . , xn) ∈ Xn , the following sets can bedefined:

X(x;>) = {y ∈ X : D(x, y) > D(x)},X(x;=) = {y ∈ X : D(x, y) = D(x)},X(x;<) = {y ∈ X : D(x, y) < D(x)} (19)

(similar definitions for X(x;≥) and X(x;≤)).


These three sets form a partition of X, with possibly one or two of them empty.

Example 7. For the Maximum multidistance DM , the sets above are

X(x;>) = X∖ n⋂

i=1

B(xi , DM (x)),

X(x;=) =n⋂

i=1

B(xi , DM (x)),

X(x;<) = ∅, (20)

where the balls on the right sides are ordinary balls in (X, d).For the Fermat multidistance DF , we have

X(x;>) = X \ Fx, X(x;=) = Fx, X(x;<) = ∅. (21)

The set X(x;<) can be non-empty, as the following example shows.

Example 8. Consider the Euclidean plane R2 with the multidistance D(x1, . . . , xn) = 1/( n2 )

∑i< j d(xi , x j ). For a list

x = (x1, x2), the set X(x;=) is the ellipse with focus at x1, x2 and semimajor axis d(x1, x2). The interior of the ellipseis X(x;<) and the exterior, X(x;>). Note that if x1 = x2 = x , then X(x;=) = {x}, X(x;<) = ∅ and X(x;>) = X \ {x}.Definition 3. Given a multidistance D and a list x = (x1, . . . , xn) ∈ ⋃

n≥1 Xn , the closed ball of center x and radiusr ∈ R is the set:

B(x, r ) = {y ∈ X : D(x, y) ≤ D(x) + r}. (22)

If n = 1 and r ≥ 0 the ball is B(x, r ) = {y ∈ X : D(x, y) ≤ D(x) + r}. As D(x) = 0, B(x, r ) is an ordinary ballin the metric space X.

Remark 3.

(1) The closed ball centered at a list x and radius 0 is the set X(x;≤).(2) There exist non-empty balls with negative radius if and only if X(x;<) �∅.

Example 9. For a given metric space (X, d), the balls corresponding to the distance DM (x) = maxi, j {d(xi , x j )} are

B(x, r ) =n⋂

i=1

B(xi , DM (x) + r ), (23)

where the balls on the right side are ordinary balls in (X, d) and r ≥ −DM (x). Let us prove it

y ∈ B(x, r ) ⇔ D(x, y) − D(x) ≤ r

⇔ D(xi , y) − D(x) ≤ r ∀i

⇔ y ∈ B(xi , DM (x) + r ) ∀i

⇔ y ∈n⋂

i=1

B(xi , DM (x) + r ).

Example 10. The balls corresponding to the Fermat drastic multidistance (see Example 3) are

B(x, r ) =

⎧⎪⎨⎪⎩∅ if r < 0,

Fx = {x ′1, . . . , x ′

k} if 0 ≤ r < 1,

X if r ≥ 1.

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Balls with negative radius are empty because DF (x, y) ≥ DF (x) always holds for any Fermat multidistance.


The element x ′l , l ≤ k, which is repeated nl = n1 times, belongs to any ball with non-negative radius, because

D(x, x ′l ) − D(x) = n + 1 − (nl + 1) − (n − n1) = 0.

On the other hand, if y /∈ Frmx then

D(x, y) − D(x) = n + 1 − n1 − (n − n1) = 1,

and so y ∈ B(x, r ) if and only if r ≥ 1.

6. Conclusions

We have proposed a formal definition of multi-argument function distance, which captures the idea of “distance forcollections of more than two points”. We think that this new tool can be useful every time you want to measure howseparated are the members of an n-dimensional list. In this paper, some general aspects of multidistances have beeninvestigated and several root examples exhibited.

Several topics and research lines arise in this new framework. For instance, aggregation of pairwise distance valuesin order to construct multidistances [6], connections between pseudo-multidistances and multi-indistinguishabilityoperators [5], asymmetric multidistances and aggregation of multidistances.

References

[1] L.M. Blumenthal, Theory and Applications of Distance Geometry, second ed., Chelsea Publishing Co, New York, 1970.[2] B. Bustos, T. Skopal, Dynamic similarity search in multimetric spaces, in: Multimedia Info Retrieval, 2006, pp. 137–146.[3] H.S.M. Coxeter, Introduction to Geometry, John Wiley& Sons, New York, 1961.[4] L. Mao, On multi-metric spaces, Scientia Magna 2 (1) (2006) 87–94.[5] J. Martín, G. Mayor, An axiomathical approach to T-multi-indistinguishabilities, in: IFSA-EUSFLAT 2009, 2009.[6] J. Martín, G. Mayor, Aggregating pairwise distance values, in: EUROFUSE 2009, 2009, pp. 147–152.[7] J. Rintanen, Distance estimates for planning in the discrete belief space, in: Planning & Scheduling, 2004, pp. 525–530.[8] E. Weiszfeld, Sur le point pour lequel la somme des distances de n points donnes est minimum, Tohoku Mathematical Journal 43 (1937)

355–386.[9] D.H. Wolpert, Metrics for more than two points at once, in: Y. Bar-Yam (Ed.), International Conference on Complex Systems ICCS 2004, Perseus

Books, in press.

Documents

Multi-argument distances