Perlodlca Mathematica Hungarica Vol. 16 (2), (1985), pp. 135--138
A VECTOR-SUM THEOREM IN TW0-DIMENSIONAL SPACE
I. B / ~ R / [ N Y ( B u d a p e s t ) a n d V. S. G R I N B E R G ( H a r k o v )
Abstract
G i v e n a finite set X o f v e c t o r s f r o m t h e uni{~ ball of t h e m a x n o r m in t h e twod i m e n s i o n a l s p a c e w h o s e s u m is zero, it is a l w a y s possible t o w r i t e X = {x 1. . . . , Xn}
k
in s u c h a w a y t h a t t h e fix'st c o o r d i n a t e s of each p a r t i a l s u m x - xl lie in [ - - l, 1 ] a n d t h e
1
s e c o n d c o o r d i n a t e s lie in [ - - C , C] w h e r e C is a u n i v e r s a l c o n s t a n t .
R a denotes, as usual, the d-dimensional Euclidean space with a fixed
basis, x j is the j-th coordinate o f x E R d. 1X] denotes the cardinality of X.
F o r the sake of b r e v i t y we write s ( X ) = Z~{x: x C X} if X c_ R d, IX] ~ ~ .
C 1, C 2. . . . will d e n o t e i n d e p e n d e n t constants.
A well-known t h e o r e m of Steinitz [4] asserts t h a t if X c B c R a,
B is c o n v e x a n d c o m p a c t , IXI --- n and s ( X ) -= 0, t h e n there exists an indexing
X ~- {Xl, x 2. . . . . xn} such t h a t
xl + x
2+...
+x k~B=
{~b~Rd:bEB}
for e a c h / c ---- 1 , . . . , n, where 2 ~ 0 depends o n l y on d a n d B.
This result was significantly i m p r o v e d in [3]: it is shown there t h a t
;~ =- d will do for a n y convex, c o m p a c t B. Using this t h e o r e m we are going
to give a c o o r d i n a t e - d e p e n d e n t b o u n d for the partial sums in two-dimensional
space. More precisely, we will p r o v e the following
THEOREM. Let
X c {x
R2: Ix1[ < 1, Ix2[ < 1}, IXl = n, s(X) = 0.
T h e n there is a n i n d e x i n g X :
[x~ + . . .
{x 1, x 2. . . . , xn} such that, for each Ic,
+ x~[ ~ l a n d [x] + . . .
+ x~l ~ C
where C is a universal constant.
A M S (MOS) subject classi]ication8 (1980). P r i m a r y 52A40; S e c o n d a r y 05A05.
Key words and phrases. B o u n d e d p a r t i a l s u m s , r e o r d e r i n g of a series.
5*
Akaddmiai Kiad6, Budapest
D. Reidel, Dordrecht
1~6
B~R-~NY, ORINBERG: A VECTOR-SUM THEOREM
I t is clear t h a t the b o u n d for the first coordinates o f the partial sums
c a n n o t be i m p r o v e d so in this sense our t h e o r e m is best possible.
W e m e n t i o n f u r t h e r t h a t our t h e o r e m will be p r o v e d with a n effective
O(u ~) algorithm, so it m a y t u r n o u t to be useful in applications o f the Steinitz
l e m m a in scheduling t h e o r y (cf. [1], [2]).
W e split the p r o o f into four steps. W e s t a r t with a definition. A set
r=
{Yl, Y2, . . ", Ym} c Ra
m
is called a bounded grouping o f X c R a if there is a p a r t i t i o n U Xi o f X
i~1
(Xt N X i : ~J for i ~ - i ) with [Xt[ ~ C 1 and Yt :
s(Xi)"
LEMMA 1. I f Z c [ - - 1 , 1], IZ] ~ 0% then it has a bounded grouping Y
whose diameter is not larger than 1.
PROOF. Adding zeros to Z if necessary, we m a y assume t h a t Z ---- Z + U
U Z - , Z + M Z - = O, Z - c [ - - 1 , 0], Z + c [0, 1] a n d IZ+I = ] Z - I = n. L e t
us order Z - - - - - { z 1,zz . . . . ,Zn} with zl ~ z 2 ~ . . . ~ z
n a n d Z + = {u 1,u 2,
• .., un} with Ul ~ us ~ • • • ~ un. Consider the p a r t i t i o n
u
u
}u . . . u
un}
o f Z. This gives a b o u n d e d grouping Y ---- {Yv • •., Y~} with Yi -~ zi + u r
The d i a m e t e r o f Y does not exceed 1 because
ly, -
yjl =
Ix, -
+
u, -
1.
LEMMA 2. I f Z ~ [ - - 2 6 , 1 - - 26] with 6 E [0, 1/4] and IZi < ~ ,
Z has a bounded grouping Y c [ - - 2 e , 1 -- 3e] with some e E [0, 1/4].
[]
then
PROOF. Define
Z----- { z E Z : - - 2 ~ z ~ - - ~ }
a n d Z + ---- { z E Z : I
-- 3 ~ z ~ l - -
2~}.
Observe t h a t z E Z - a n d u E Z + i m p l y -- 6 ~ z + u ~ 1 -- 36. F o r m disjoint
pairs {z, u} with z E Z - and u E Z + until possible - - this will be the p a r t i t i o n
for Y plus the non-paired elements o f Z t a k e n as singletons. T h e n Y is a b o u n d e d
grouping a n d if ]Z-] ~ [Z + ], t h e n Y c [--6, 1 -- 2~] so we m a y take e --~ 6/2
and if IZ-I ~ [Z + I, t h e n Y c [--26, 1 -- 36] so e = 6 will do.
[]
LEMMA 3. Z C [ - - 1 , 1 ] , IZi----n, [ Y I ~ I a n d Is(Z) + y [ ~ l .
Then
there is an indexing • = {z1, z ~ , . . . , Zn} such that [y + z~ + z z + . . . + zkl ~ 1,
(k =
1, 2 . . . .
, n).
PROOF. F o r n ~-- 1 this is evident. To use i n d u c t i o n for n ~ 1 it is sufficient to find z o ~ Z with Iz0 + y] ~ 1. Assume y ~ 0 (the o t h e r case is sym-
137
B ~ R ~ N Y , G R I N B E R G : A VECTOR-SUM T H E O R E M
metric). T h e n one can take a n y non-positive element o f Z for z o, a n d i f
each e l e m e n t o f Z is positive, t h e n one can take a n y of t h e m for z 0 because
z o+y~s(Z)
+y~l.
[]
LE~r~A 4. Let
X = {x 6 R 2: Ix~] :~ 1, I~1 < a}, IXl =
a n d Y = {Ya . . . . , Ym} be a bounded grouping o/ X with [Yl + . . . + Ylk[ ~ 1
a n d ly~ + . . . + y~] ~ V 1 (k = 1 , . . . , m). T h e n there is an indexing of X =
:
{Xl, X 2 , . . . , Xn} such that Ix I + . . . + x~[ ~ 1 a n d [x~ + . . . + x~[ ~ C~
(It = ] , . . . ,
n).
PROOF. X will be ordered in the same w a y as Y , so we h a v e to give the
ordering inside the groups Xi where Yi = s(Xi). To do so we a p p l y L e m m a 3
for the first coordinates. This shows t h a t the first coordinates of each partial
sum lie in [ - - 1 , 1]. B o u n d e d n e s s o f the second coordinates o f the partial
sums follows from the b o u n d e d n e s s o f the grouping.
[]
PROOF o f the Theorem. First, a p p l y L e m m a 1 for the first coordinates
o f X. F o r s y m m e t r y reasons, we m a y suppose t h a t the b o u n d e d grouping
we get lies in [--2~, 1 -- 23] for some ~ E [0, 1]4]. Applying L e m m a 2, we
get a b o u n d e d grouping
r c B = {x C R2: - - ~
~ x 1 ~ 1 -- 3~, Ix21 ~ C,}
with some e E [0, 1/4]. Using the result f r o m [3] m e n t i o n e d in the introduction, we find a n indexing Y = {Yl, Y2, • •., Ym} with Yl + g2 + • • • + Yk C 2 B
(k = 1, 2 . . . . , m).
N o w we are going to find a n indexing for Y such t h a t the first coordinates o f each partial sum lie in [-- 1, 1 ]. I f this is n o r s e for the indexing Yl,
Y2 . . . . ,Ym, t h e n e ~ l / 6
a n d y] + y~ + . . . + yp1 ~ 1 for some p. T h e n
8(Y) = 0 a n d - - 2 e < y] i m p l y the existence o f q ~ p with 1 -- 6s ~ y] +
+ . . . yql ~ 1 - - 4e. Set now zi = Yi+q (i + q is m e a n t modulo m). T h e n for
the indexing Y = {Zl, z 2. . . . . Zm} we have
[Zl~ + . . .
+z~[~l
a n d ]z~ + . . .
+z~[~G
2
(/~ - - 1 . . . . . m). To finish the p r o o f one has to a p p l y L e m m a 4.
[]
Carrying o u t the calculations one gets t h a t the T h e o r e m holds with
C : 18. We t h i n k this b o u n d is n o t sharp. W e m e n t i o n finally t h a t the same
m e t h o d works in R d (d ~ 2) a n d gives the following result. I f
x c {~ ~ R ~- [~1 ~: 1 , . . . ,
I~l ~ 1}, IX[ : ~ and s(X) = 0,
138
B/[RJ[NY, GRINBERG:A VECTOR-SUMTHEOREM
t h e n t h e r e is a n i n d e x i n g X = (x 1. . . . .
Xn} s u c h
that
d
[xXx ÷ . . .
(i=2,...,d;k=l
+ 4 ] <C 2 a n d [4 + . . . 4 1
. . . . . n).
REFERENCES
[1] I. B~R/~Y, A vecbor-sum theorem a n d its application to improving flow shop guarantees, Math. Oper. Ree. 6 (1981), 445--452. M R 82i: 90057
[2] I. S. BELOV a n d JA. N. STOLIN, Algoritm V odnomar~rutno~ zadaSe kalendarnogo
planirovanija (An algorithm for the single-route scheduling problem), Matemati~eska]a ~konomika i funkcional'ny~ analiz, Nauka, Moskva, 1974; 248--257.
M R 56:17780
[3] V. S. GRZ~ERG and S. V. SEVAST'J&~OV, O veliSine k o n s t a n t y Steiniea (On the
value of the Steinitz constant), Funkcional. Anal. i Prilo~en. 14/2 (1980), 56--57.
M R 81h: 52008
[4] E. STEX~XTZ, Bedingt konvorgento Reihen u n d konvexe Systeme, J. Reine Angew.
Math. 143 (1913), 128--175.
(Reoeived Jul/y d, 1983)
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