PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 35, Number 2, October 1972
SUMS OF DISTANCES BETWEEN POINTS
ON 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\l
i<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)
g(x, y) = —-——-;
^-.
x(v — v) + v(u — X)
« - 1
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
547
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Society 1972
548
[October
K. B. STOLARSKY
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-lATheorem 1.
Let usiv. Define
fiu, v) = 1,
u = v,
= (w — l)/t;,
u > t>.
Then
(3)
2 + 2^/(».»)212
12
Moreover the constant fiu, v) is best possible.
Proof.
For any two points pi,p2e
Em,
IPi - 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)^ix— yf and for v+2^u
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
l^i,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
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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^/ drv
J
Clearly
(6)
i.j
/ = 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 \(p[ - (Tl - q¡)') ■r2p0\ dr2
where bm. depends only on the dimension m . Hence by Theorem 1,
I = fffrm' 2 \P'i • T2Po- {rHÎ)' • T2Pol¿Tgi/Tj
(7)
^/_1íf ( 2 6m.|(PÍ- pí)■T^ol
JJ Xi<i
+ 2 i»m-I{(t,^)'
!<i
.
- (T^i)'} • T2p0\) dT2 (¿Ti
!
= 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,
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Urbana,
Illinois
61801