NATIONAL BUREAU OF STANDARDS REPORT

1902

ON FEJER SETS IN LINEAR AND SPHERICAL SPACES

ty

Tc, S 0 Motskln

University of California 9 Los Angeles

and

Io Jo Schoenberg

University of California , Los Angeles

and

University of Pennsylvania

U. S. DEPARTMENT OF COMMERCE

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NBS PROJECT

August .2$ s 1952

NBS REPORT

1902

ON FEJER SETS IN LINEAR AND SPHERICAL SPACES*

**

toy

To Sc Motzkin

University of California, Los Angeles

and

lo Jo Schoenberg

University of California, Los Angeles

and

University of Pennsylvania

PREPRINT

Presented to the American Mathematical Society , September 1952 .

This work was performed on a National Bureau of Standards con- tract with the University of Californiag Los Angeles,, and was sponsored (in part) by the Office of Scientific Research, USAFo

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

This report is issued for : in any form, either in wholi from the Office of the Dire<

reprinting, or reproduction

lion in writing is obtained

igton 25, D. C.

INTRODUCTION

lo Let M be a metric space of points asb 9 «°°

, with the distance from a to b denoted by abo Let A be a given subset of M„ If

p and p® are points of M such that

(1) px > p

9 x , for every point x A A ^

we say that p

9 is point-wise closer than p to the set A, We say

that p is a point-wise closest point to the set A if no point p

exists which is point“Wise closer than p to A„ For brevity we shall

also refer to a point-wise closest point to the set A as a minimal

point of the set A„ The set of minimal points of A will be called

the Fejer se_t of A and denoted by F(A) C This set is never void

since F(A) o A. F(A) is a monotone set function since

(2) A

1

C A. implies l{\) £ nK) .

Let p a p

9 be distinct points of M0 The set of points x such

that

px i p

9 x

is called a half-space of M and denoted by the symbol H - HCp^p*)!

p s p

are its defining points or foci , p being exterior to H s p

9

interior o The convex hull K(A) of a given set A is defined as

follows: If A is not contained in any half-space 9 we set K(A) -M.

If there exist half-spaces containing A s then K(A) is defined as

the intersection of all such half-spaces:

( 3 ) K(A) -OH.

H 3A

" 2

"

Notice that K(A) is always closed and A g

K(A). A further general

inclusion is

Indeed,, if p £ F(A ) 9 then a point p* exists such that (1) holds 0

This implies A £ H - H(p s p

! ) while evidently p l H„ By ( 3 ) p

e K(A)

and (i|) is established.

2 o In 1922 Fej er [ lsj

s noticed the interesting fact that in

the euclidean plane M " E,, we have

One half of Fejer’s own argument has just been used to establish

(It) for any metric space „ The other half of Ms proof in E0 is as

follows « Let us show that F(A) c K(a)

,

Indeed s if p e K(A) S then

there exists a half“plane H c E.; such that p €. H } A c H„ If p

5 is

the point of H which is nearest to then (l) evidently holds

showing that p £ F(A)

.

The last paragraph allows of a wider setting „ Let M be a

real inner“product space 0 By this we mean that M is a real, linear

space whose norm springs from an inner product (a s b) and that M is

complete with respect to the metric just describedo Under these

assumptions the convex hull K(A) is found to be identical with the

least closed and linearly connected set containing A 0 Fejer 3 s argument applies unchanged to prove the following

(W K(A) C . F(A) .

(*> F(A) - K(A) ,

then

(6) F(A) - K(A) .
'
-3~

3c In Parts I and II "we investigate the Fejer sets of finite

sets of points in Banach spaces. The notion of the global distribution of points on spheres is introduced (Definition 1) and Fejer sets

are described in terms of this concept (Theorem 50° Fejer 9 s result

is shown to hold in rather arbitrary 2-dimensional Banach spaces

(Theorem 6), The situation is quite different for dimensions exceeding twOo A very weak form (2d) of Fejer 9 s result is shown to imply

that the Banach space is an inner-product space 0 In Part III we determine the Fejer set of a subset of spherical space.

I. ON FEJER SETS OF FINITE SETS OF

POINTS IN BANACH SPACES

in An application of Kelly9 s theorem . Let B be a Banach space

to be also denoted by B.,. in case its dimension n happens to be finite. The Fejer set FCa^b) of two points is described by

THEOREM 2, In any space B we have that

(Id) F(a,b) - E(x| |ja-xjj + ||x-b|| - ||a~b||) .

Proofs Indeed^ a point x of E is clearly minimal for the set

| a s b 'because the existence of a point x 9 point-wise closer than x

to (a s bj would violate the triangle inequality. On the other hand,

if i|a-xf + ||x~b|| > Jja-bf 8 we can find a point x

9 on the segment

joining a to b such that jja-xll > j{a”X !

|j 9 ||b-x(j > ||b-x 9 |jo This

proves (ld)o

THEOREM 3o Let A be a finite set of points of B (n finite)

and let us assume that A contains at least n+1 points . Then

(1°2) F(A) - UF(poS Pl , (P1

os>o P ) ^ A), 9 »

where the elements of the union are formed for ere ry combination

of n+1 distinct points of A„

Proof ; Let 6(A) denote the set on the right-hand side of

(1,2). By (2) we have

(1»3) 6(A) c F(A) .

Assume now that

(loh) p £ 6(A)

and let us show that p “ F(A), i.e. a that p is not minimal . For

every point a t A we consider the open sphere

(lo^) Sa

: |x~a|| < ||p-a| .

Let pQS p be any distinct points of A, By (i 0 ii)

P £ *(P0

,

a ”,Pn

)* This means that a point-wise closer point exists

for the set ; d p or

c

s 9 i n

5

( 1 , 6 ) s n s n

Pi p.

The spheres S are convex sets, finite iri number9 every n+1 of which

have a common point , in view of (1,6). By Helly’s theorem (see [ 5] )

n

a e A

Sa f 0 o a

If p

J % H S then p

9 t S for every a, or |p°a.j! > |jp*=&!| for

every a £ A. Hence p I F(A) 0 Thus 6(A) 3 F(A) . In view of (l 0 3) s the identity (l 0 2) is established.

An example o Let ~ be the Minkowski space of points x " (x. , 000

s 3^) with the n cubical” norm ||x|[ = max|x<,jo In this A

i

1

particular case the resul Theorem 2 maybe improved as follows?

=

5

“

If A is a finite set of points in M._. then

( 1 . 7 ) 7(a) - u y(p

o

, Pl > ,

where the union is formed for all pairs of distinct points of A0

Indeed, in this special case the spheres (l 0 £) are open cubes

with edges parallel to the axes of coordinates . On repeating the

argument used in proving Theorem 3 we find that every two among

the cubes have a common point,, If follows that the projections of

the cubes on each coordinate axis have a common point and that

therefore also all cubes have a common point

„

Let n = 3 and let A - fas b 3 c], where

(108) a - (2, 0, 0), b - (0, 2, 0), c - (0, 0 3 2) .

By (lc7) we have

F(as h g c) - T(a, h) + F(a9 c) + F(b s c) .

The inspection of a diagram will show that the three right-hand

side sets are lozenges: F(a9 b) is the lozenge of consecutive vertices (2 S 0 g 0), (l, 1, l) s (0, 2 S 0), (l, 1, “1). Notice in particular the curious fact that the centroid (2/3, 2/3, 2/3) of the

triangle A(a s b 9 c) is not a point of the Fejer set F(a s b s c)

.

However the point p

~ (l, 1, 1) does belong to T(a s b 3 c). We

shall say that a 9 b,c are globally distributed on the sphere fx-pj!"!

and investigate the general concept in our next section.

On points globally distributed on spheres and their characterization. DEFINITION 1. Let B be a Banach space and let

(109) S : ||x=c|| s r ,

be the surface of the sphere of radius r and center at c„ Let

, Pk k® k of S. We shall say that the point

Fl

9

( 1 . 10 )

are globally distributed on S

c e F(p , p.

• V •

In other words ; There is no solid sphere ||x-c 3 6 of a smaller

radius r* < r a which covers all Pn k*

In what follows we denote by [a,b) the half-open segment of

points a(l-t) + bt (0 - t < 1), joining a with b s and use similar

notations [a # b], (a s b) s for closed or open segments.

DEFINITION 2 . Let p^ ,

°

' 3 be k points on the sphere

(1.9) of Definition 1. We say that the points p^ ,

• •

9 p^ are well

visible provided there exists a point q outside S, j] q—c || > r f such

that the segments [q 9 p/) contain only points exterior to S and that

there are points s^ s such that p1 e (q s s^) and that the segments

(p^ 9 s^] have only points interior to S.

Example . In E0 k points on a circle ||x=c|| = r are well visible

if and only if they are contained in an open half“circle. The general relation between Definitions 1 and 2 is shown by the following

THEOREM ho The points p dis- n 9 p9 ,

° »

° 9 p,. on S are globaj

I

s ^2 > s tributed on S if and only if they are not well visible .

Proof ; a. The condition is necessary . Indeed, suppose

that our condition is not satisfied and, on the contrary, the points

p^ are well visible from the outside point q. By Definition 2 we

can extend the segment [q 9 p ] by a segment (p^,s,] all points of

which are interior to S. Let us now "shrink” S from the center of

similitude q in the ratio 1:A(0<A<1), obtaining a new sphere

-

7

~

S' of radius r* = rX < r. Let the segment (p^, s^) be transformed

into (p| s s^) by this similitude transformation. Since (p. s s^)

is interior to S (p! si) will be interior to S’. It is clear that

X X

pi

e (pi* s

i) » (i “ 1, *•*

, k ),

provided that A is sufficiently close to unity. But then all p^ are

inside S', showing that p^ s

•

• • a p^. are not globally distributed on

S.

b. The condition is sufficient . Indeed, let us assume that p^ # p^ are not globally distributed on S and let us

conclude that they are well visible. Accordingly, let p^ be covered

by a sphere of radius r^ < r. Inflate S^ slightly from its center into a larger sphere of radius r* < r. Now all points p^ are

interior to S. Let c* be the center of the sphere S’. Since the

p^ are on S and inside S’ s we must have c’ f c. The spheres S and

S® are similar with respect to the (exterior) center of similitude

re'-r'c

r-r 1

We denote by p" = Tp the similitude transformation with center at

q and ratio r Let p. - Tp„ 0 Since p. is on S, p. is on S

But p is by construction inside S’. It follows that all po:

of (p^, p^] are inside S’. Let s. be such that

p. = Ts ol

Now (p s^] goes over into (p^ s p^] by our transformation. Since

(p^s P^l was shown to be inside S® s it follows that (p^, s^] has

only points interior to S. We now claim that all points of [q, p^)

8-

are outside So For if any point of this segment were inside or on

S s the fact that s^ is inside S would imply that also p^ were inside S s which is not the case. We have just shown that [q p. ) is

i

outside S s (p^, s^] is inside S a (i - l s *°°

9 k) 0 But this is precisely what we mean when we say that the points pi are well visible

from q.

6. A description of Fe.j er sets in terms of global distribution.

THEOREM Let A = j p^s

• •

• $ p^j be a finite set of points

of B. Let p £ A. Draw about p as center a sphere

S s ||x-p|| = r

and project p^ from p onto the surface S into q^ a Denote by F-^(A)

the set of those points p such that the points q^ , , q^. are

globally distributed on S. Then

(loll) F(A) - A + 7 (A) .

Proof ; We have to show that p F(A) if and only if the q^

are globally distributed on S or equivalently:

(1.12) p T F(A)

if and only if

(loll) q^ ? q^ are not globally distributed on S.

The size of the radius r > 0 is clearly immaterial.

a. Let us assume ( 1 . 13 ) and prove (1.12). Choose r such

r < min ||p-pi

|| .

that

-9-

The assumption (l»13) means that pCF(q^ s

»

• 3 q^). Therefore there

exists a point p* such that

lhL -p|| > Iki-P' II , (i - 1, •••

k).

But then

IIp^pII = llp^il +

llq^-pl > + llq^p’ll - IIp^p ' II or

IIp^pII > Ip^p'II .

which proves (l„12)„

bo We now assume that (l„12) holds and wish to prove

(I 0 I3 ). Choose r such that

r > max Ip-pJ . i

By (l 0 12) there is a point p* such that

IIp-pJI > IIp'-pJI , for all i.

Hence

r - ||p-qi

l - ||p"Pi

!| +

l|p i“q

i

l| > flp'-pjl + llPj-qJI

~ Ip'-qJI or

Up’-qJI < r , for all i.

Setting

r

! = max fp’-q. || < r

i

we see that the smaller sphere ||x-p’|] - r' will cover all points q^,

which proves (I 0 I3 ).

7« £2 Fe-i /&v se ^ s in B^o As an application of Theorem b we

wish to describe the Fejer sets of finite sets of points in a

-10-

Minkowski plane B^. By Theorem 2 we conclude the following: The

Fejer set F(a,b) is identical with the segment [a,!*], for every pair

of points a,b, if and only if the norm of B^ has the property:

IIpII + h\\ =

l!p +

ql! , p ? o , q f o (1.1U)

imply that p

= aq , (a. > 0) .

It is known that (l 0 lb) holds if and only if the gauge- curve ||x||=l

contains no segment, in which case the curve ||x||-l may be described

as being "round .

We assume (l„lb) to hold and consider in B

? a set A = jp^jP^jP^)

of three points. Let p£A and let be the points on the

gauge-curve S as described in Theorem h. By Theorem 5> p £ FCp^p^p^)

if and only if q^q^q., are not globally distributed on S B By

Theorem b this is the case if and only if q^q^q^ are well visible

(from a point q outside S)„ S being "round," this is evidently the

case if and only if p is outside the closed triangle A(q^ s q^)

or, equivalently, outside the triangle A(p^, p^, p^). Thus

F(p1? P2

, P

3

) = A(p

1

, p2, p

3

) .

Applying Theorem 3 we obtain

THEOREM 6. Let be a Minkowski plane with the property

(l.lb). The Fej er set of a finite set of points is identical with

the closed convex polygon spanned by the set .

This result is a special property of 2-dimensional Banach

spaces as will be shown in Part II.

-11-

II. A CHARACTERIZATION OE INNER-PRODUCT SPACES

8 . The main theorem . Let M be the inner-product space of

Theorem 1 and let the set A consist of three points f p^ , p^, p^

distinct or not. Then K(A) S being identical with the least closed

and linearly connected set containing these points, is evidently

the triangle A(pn 9 p„ 9 p.) having as vertices the points P-.jPnsP^*

By Theorem 1 we have

(2*1) ? (P1S P2

, P

3

) s A (P1S P2s P

3

) •

Let now Pi be an arbitrary Banach space 0 By Theorem 6 we know

that (2d) again holds provided that B - B

? is 2-dimensional and

that its metric has the property (l.lU). Which higher-dimensional

Banach spaces B enjoy the property (2„l)? An answer is given by

the following

THEOREM 7 o A space B of dimension - 3 has the property (2.1)

if and only if it is an inner-product space .

9. A few lemmas . Eor the proof of Theorem 7 we need a number of results concerning the 3 “dimensional euclidean space E^.

LEMMA I. Let denote the surface of a sphere in E^ with

center at o. Let C be a simple closed curve on and let H^ and

and H^ be the two open components of the complementary part ; =

C + H^ + Hg. Let K*(H^) and K*(H

2

) denote the least convex sets

in containing H^ and E^ s respectively. If

( 2 . 2 ) oeK*^) U k*(h

2

)

then C is necessarily a great circle .

- 12-

Proof ; By (2.2) o€.K*(H^). Therefore o is either outside

or on the boundary of the closed convex set

K* (H^) = K*^ + C) 0

We can therefore draw through o a plane TT which is a bounding

plane or a plane of support of K*(H^ + C) . Let P = fl IT and let

h^ and h^ denote the open half-spheres in which IT divides ;

^2 ” P + hp + ^

2

° our choice ^ and ^or a proper labelling

of the h„ we have

1

h

+ r $ h

•

Passing to complements on the sphere we see that h^ C + C, and

since h^ is open on the sphere, we have

(2.3) h

2 C h2

. We now claim that

(2.U) n h

2

= 0 .

Indeed, if p eh^ fl H

2

, then K*(p, hg) would contain the point o and

by p

a H

2

and (2 0 3 ) a fortiori

oeK*(H

2

) ,

in contradiction to (2.2). By (2.3) no point of C is in h

2 and by

(2.U) no point of C is in h-^„ Thus C c f and therefore C = P

showing that C is a great circle.

A seemingly more general version of Lemma 1 to be used below

is as follows.

LEMMA 2 . Let S be the surface of a convex body in E, having

the point o as center of symmetry . Let C be a simple closed curve

-13"

on S and let H^H,, be the two ojoen cOTEone^ °£ iM SSS^S^:

s = c + ^

+ H

2

. Let K*(H1

).K*(H

2

) denote the least convex sets

in E containing E,_ and Hg, resEeotivels- If

otK*^) U K*(H

2

)

. , „ ^

q A e C is the intersection of S

then C is a "great circle of S, i ... 9

b£ a plane through o.

Proof ; Draw a sphere S

2

with center at o. Central pr

jection of S only S

2

with the center of projection at o reduces

Lemma 2 to Lemma 1«

We turn to our last le«na which is due to Blaschke. Let S

be the surface of a convex body in Ej. We assume that the surface S contains no se^ent, a fact which we describe by saying

that S is "round." Let * he a fixed vector and let us illuminate

S by rays of light all parallel to * in direction and sense. Let

, , _ o +hat is the "boundary beC denote the light-shade boundary on S. that is,

. 4. e aTVa the part of S which is shaded,

tween the illuminated part of S and P

We claim that C is a sirffile closed curve. Indeed, let * be a

fixed plane normal to v. The orthogonal projection of S onto T is

a plane convex domain D whose boundary curve we den 7 P

Let p

e T • The point p is the projection of a point q of S. Since

S contains no segment this point q is unique and is readily shown

t0 vary continuously as p varies continuously on P- As p de-

.

scribes P the point q describes our curve C which is clearly simpie and closed.

LEMMA 3 (Blaschke). If S has the H£E2£iZ Sat &S 1^'

nfO*
V
shade boundary C is a plane curve for all possible directions of v

then S is necessarily an ellipsoid .

Blaschke proves this result ([l], pp 0 l£7=l£9) hy assuming

that S is an "Eiflache," by which he means that S is analytic and

regular at all its points and has everywhere non-vanishing curvature ( [ 1 ] 3 p 0 llj.7). However 9 the reader will have no difficulty in

carrying through Blaschke "s proof on the basis of our simplified

assumption that S is "round 0 H

10° Proof of Theorem ?• We are to show that the property

( 2 o 1 ) implies that B is an inner-product space,, If p-, - p^

then

(2d) implies that

F(P1S P2

) s tpls P21

.

The metric of B must therefore have the property (I0II4): The

gauge-surface

2 : ||x|| - 1

of B is "round

Let B^ be an arbitrary but fixed 3

=dimensional linear subspace of B 0 B^ is a 3"dimensional Banach space whose gauge-surface is

s = b n 2 . 3

In terms of a coordinate system in B we can also think of B as

3 3

being a Minkowski space whose points are those of an E^ which is

metrized by means of the convex gauge- surface S. By a theorem of

Jordan and von Neumann [3], it suffices to show that S is an ellipsoid.

We know already that S is "round" because 2 has this property

(S C Z and Z contains no segments). Let us illuminate S from a direction parallel to v. Let C be the light- shade boundary on S and

let be the illuminated part of S, the shaded part, 3 = 0 + ]^+ H

We claim that

(2o 50 ofeK*^) .

Indeed, let us assume for the moment o to be a point of the

convex set K*(H^)„ being connected it follows, by a sharpened

. 1 version of a theorem of Fenchel , that o is the centroid of some

three points of Pp s P2

* P

3

> say. Thus

(

2

» 6 ) Pls P2

, P

3

eH

i

»

(2-7) o e a(p13 p2

, p

3

) .

These conclusions, however, are contradictory with our previous

assumptions, for on the one hand the points p^, p^, p^ are well

I n^l visible from a point at infinity in the direction -v. Prom this

it follows easily that they are well visible from a point q at finote distance and sufficiently far out in the direction of -v. By

Theorem h we conclude that the points pn 9 p~, p are not globally

± z 3

distributed on the S of the B_

On the other hand by (2„7) and our basic assumption (2d) we

conclude that ofeP(p^, p2

, p^). Thus o is a minimal point of the

set A - [p p?J p ] in the space B 0 It follows a fortiori that

i The sharpened version of the theorem of Fenchel is as follows:

Let H be a connected subset of and let K* (H) be the least con- vex set containing H.~~ Then every point of K*(H) is a centroid of some n points of H. This result is due to L. N. H. Bunt. See [2]

for Bunt's proof (pages 5>89"5>90) and for references.

-16-

that o is minimal point of A with respect to the subspace B . Thus

P2

* P

3 are globally distributed on S in B in direct contradiction to the conclusion of our previous paragraph.

This proves (2 a 5) and we may similarly show that

olK*(H

2

) .

By Lemma 2 we conclude that C is a plane curve and by Blaschke’s

Lemma 3 we learn that S is an ellipsoid.

III. ON FEJER SETS IN SPHERICAL SPACES

11. Let M be the real inner-product space of section 2. We

are now confining our attention to the surface S of its unit sphere

||X|| = 1 o

By the distance xy of two points of S we mean the arc defined by

cos xy = (x,y) , o ^ xy ^ ir

If p and p’ are distinct points of S then the closed half-sphere

H(p, p>) may be defined by the inequality px - p

s x or equivalently

by (p s x) - (p' s x) or (p'"p s x) - 0. A being a given subset of S

we may now define the convex hull K(A) as in section 1.

THEOREM 8. Let A be a subset of S. a. If there is no open

half- sphere containing A then

(3.1) F(A) = S .

b. If there is an open half- sphere H

q

such that A c HqJ

then

(3.2) E(A) = K(A)

-17-

Proof: a. In order to prove (3°l) we have to show that

every point p e S is a minimal point of S„ This is clear s for otherwise there would exist a point p® such that (l) holds. However (1)

implies that A is in the open half-sphere H (p s p®) which contradicts our assumption,

b. Let us assume that

(3.3) A c H

o ,

where is an open half- sphere defined by

(3.U) H

o

: (x,b) > 0 ,

and let us prove (3.2). However, the inclusion

K(A) c F(A) ,

or (U), has already been established In section 1 for any metric

space. There remains to show that

(3.5) F(A) c K(A) ,

or that

(3o6) p£K(A)

implies

(3.7) p & f(a) .

Assuming (3° 6) means that there is a closed half-sphere H such that

(3.8) ptH , H D A 0

Let the half- sphere H be defined by

(3.9) H : ( xs a ) - 0 *

By the first relation ( 3 . 8 ) we know that (p,a) < 0. Choose e > 0

-18-

so small as to make sure that

(3-10) (p,a) + e(p,b) < 0 0

Consider the open half- sphere

(3-11) H

q

: (x,a) + e(x s b) > 0 .

If xtA then xfe-H, by (3,8), and xfcH

Q

, by (3.3)° The point x thus

satisfies the inequalities (3«h) s (3.9) and therefore also (3.11):

(3.12) ACH’ .

On the other hand (3.10) shows that

pen; .

But then clearly p is not a minimal point of A. Indeed let p* be

the symmetric of p with respect to the hyperplane

(x, a+fcb) = 0

which bounds H^„ Clearly - H

o

(p s p’)» By (3.12) we see that p*

is point-wise closer than p to A, This proves (3-7) and therefore

also our theorem. Notice that the dimensionality of the space M,

finite, denumerable or non-denumerable, does not affect our theorem.

August 28 9 195>2

-19-

REFERENCES

[1] Wo Blaschke, Kreis und Kugel , Leipzig , 1916.

[2] 0. Hanner and H. Radstrom, A generalization of a theorem of

Eenchel , Proceedings of the American Mathematical Society, 2

(1951), 589-593-

[ 3 ] P. Jordan and J. von Neumann, On inner products in linear

metric spaces . Annals of Mathematics ( 2 ), 36 (1935), 719“723.

[ U] L. Fejer, liber die Lage der Nullstellen von Polynomen , die

aus Minimumforderungen gewisser Arten entspringen . Math.

Annalen 89 (192

) s I4I-I4.8.

[5] Ho Rademacher and I„ J 0 Schoenberg, Convex domains and

Tchebycheff j s approximation problem ,, Canadian Journal of

Mathematics 2 (1950), 2h5"256.

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