sums of distances between points on a sphere

PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 35, Number 2, October 1972

SUMS OF DISTANCES BETWEEN POINTSON A SPHERE

KENNETH B. STOLARSKY1

Abstract. An upper bound for the sum of the Ath powers of all

distances determined by N points on a unit sphere is given for

Let plt ■ • •, pN be points on the unit sphere UmofE'", the w-dimensional

Euclidean space. Let

S = S(N, m, X) = max 2 \P< - Pj\li<i

where the maximum is taken over all possible/*!, • • • ,/*v. The function

S has been studied in [l]-[5]. Let a(Um) denote the surface area of

Um. Björck has shown in [3] that S<c(m, X)N2 where c(m, X)=

\a(Um)~l J" \p0— p\* do(p); here j" • • • do(p) denotes a surface integral over

Um and /*„ is an arbitrary point on Um. For example, c(3, 1)=|. In [1]

Alexander used Haar integrals to prove

(1) S(N, 3, 1) < IN2 - I

By the same technique we will show (see comment after Theorem 2) that

forO^A^l,

(2) S(N, m, X) = c(m, X)N2 - \c(m, X)

is true at least "half of the time"; more precisely, for any positive integer

n the inequality (2) is true either for N=n or for N=n+1. Our proof is

based on a new generalization of the triangle inequality, Theorem 1, and

an imbedding theorem of I. J. Schoenberg [6] or [7, p. 527]. Note that

when m=3 and X=l our result is weaker than Alexander's.

Lemma 1. Let u=v + 2~='i and let H denote the hexagonal region

O^x^w, O^y—^v, 1—x+y^u+v— 1. Then for (x, y) in H we have

x(u - x) + y(v - y) « - 1g(x, y) = —-——-; ^-.

x(v — v) + v(u — X) V

Received by the editors December 29, 1971.

AMS 1970 subject classifications. Primary 52A25, 52A40.

1 Research supported in part by the National Science Foundation under contract

G P-12363.© American Mathematical Society 1972

547

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

548 K. B. STOLARSKY [October

Proof. Let s denote the substitution which takes x and y to u—x and

v—y respectively. The function gix, y) is invariant under s, so it suffices to

prove the inequality for the three critical points iliu+v), Hu+v)),

Uiu+v), i(u+v)), (Jw, \v), and the three boundary lines x+y=l, y=0

and x=u. The straightforward details are omitted.

Now let Si be a finite set of u points P\," ' ,p„ in Em and similarly let

S2 be a set of v such points qx, ■ ■ ■ ,qv. Let £i -2<<j I/>*-/>* I. £2 =

Lo l?<-?il. and 2i2 =Ii.i \Pi-lA-

Theorem 1. Let usiv. Define

fiu, v) = 1, u = v,

= (w — l)/t;, u > t>.

Then

(3) 2 + 2^/(».»)2-12 12

Moreover the constant fiu, v) is best possible.

Proof. For any two points pi,p2e Em,

I Pi - Pal = am |p, • p - p2 ■ p\ doip)

where am depends only on the dimension m. Hence it suffices to prove (3)

in the case where the/;, and c/, are real numbers. Let J be any interval on the

real axis which does not contain any point pt or q¡. Let x be the number of

pt to the left of y and y the number of q¡ to the left of/. It suffices to show

L^fiu, v)R where L is the number of times J is counted on the left of (3)

and R is the number of times it is counted on the right, i.e. that

x(u - x) + y(v - y) £f(u, v){xiv - y) + yiu - x)}.

For v^ u^v+l this follows from (x— y)iu—v)îx— yf and for v+2û

it follows from Lemma 1. Equality holds in (3) if (mo/ only if)p2=/>3=

• • ■=pu=qï = - ■ ■=qv and Pi is arbitrary.

Theorem 2. For O^X^l,

(4) S(u. m, X) + Siv, m, X) <; 2c(/w, X)uvfiu, v).

Proof. Fix /. Given any / points plt ■ ■ ■ ,pt in Em, we can find an

m'^.m and / points p[, ■ • ■ ,p't in Em' such that \p'i—p'i\:=\Pi—pj\x for

lî,j^t. This result is due to I. J. Schoenberg [7, p. 527]. In the following

t, rlt r2 shall denote elements of the special orthogonal group SOim)

acting on Um. The symbol f • • • dr shall denote a Haar integral over that

group; we normalize the measure so that jsotm) dr=l. Choose/Jj, • • • ,pu


1972] SUMS OF DISTANCES BETWEEN POINTS ON A SPHERE 549

and qu • ■ ■ , qv on Um so that

(5) S(u, m, X) = 2 \Pi - p3\* and S(v, m, X) = 2 \qt - q¡\k.

Define

1= \ 2 \p,i- n^/ drvJ i.j

Clearly

(6) / = uv l\p - TV p\xdTl = uvdU"1)-1 (\Po - p\xddp) = 2c(m,X)uv

On the other hand, for any p0 on Um',

\Pi-r- q,]* = \p'i - (r^,)'! = bm. I \(p[ - (Tl - q¡)') ■ r2p0\ dr2

where bm. depends only on the dimension m . Hence by Theorem 1,

I = fffrm' 2 \P'i • T2Po - {rHÎ)' • T2Pol ¿Tg i/Tj

^/_1íf ( 2 6m. |(PÍ - pí) ■ T^ol(7) JJ Xi<i .

+ 2 i»m- I {(t,^)' - (T^i)'} • T2p0\ ) dT2 (¿Ti! < i !

= f-1 I (2 \pt - p,\X + 2 19, - qA drv

The result follows from (5), (6), and (7).

To obtain (2) set v=u— 1 in (4) and note that cu2—^c+c(u— I)2—

\c=2cu(u—l).

References

1. R. Alexander, On the set of distances determined by n points on the 2-sphere (in

preparation).

2. R. Alexander and K. B. Stolarsky, Some extremal problems of distance geometry

(in preparation).

3. G. Björck, Distributions of positive mass which maximize a certain generalized

energy integral, Ark. Mat. 3 (1955), 255-269. MR 17. 1198.

4. L. Fejes Tóth, On the sum of distances determined by a pointset. Acta Math. Acad.

Sei. Hungar. 7 (1956), 397-401. MR 21 #5937.

5. G. Pólya and G. Szegö, Über den transfiniten Durchmesser (Kapazitätskonstante)

von ebenen und räumlichen Punktmengen, i. Reine Angew. Math. 165 (1931), 4-49.

6. I.J. Schoenberg, On certain metric spaces arising from Euclidean spaces by change

of metric and their imbedding in Hilbert space, Ann. of Math. 38 (1937), 787-793.

7. -, Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44

(1938), 522-536.

Department of Mathematics, University of Illinois, Urbana, Illinois 61801


sums of distances between points on a sphere

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