Proof of the theorem

2 JAI,

5.4 Proof of the theorem

Let A 1, A2, and A be subsets of V. The set of midpoints of A I and A2 is the set

mid(A1,A2)- a, +a2

1

2

Iai EA1,a2EA21.

If K is convex and A 1, A2 c K, then mid(A1, A2) C K. Let

mid(A)-ta2a Ia,a'EA}

denote the set of midpoints of A. Then A c mid(A)

and

I2A I - Imid(A)I.

Lemma 5.11 Let V be an n-dimensional Euclidean space. Let B be a block in V, and let W c int(B). Then

mid(W, vert(B)) c int(B)

and

I mid(W, vert(B)) I - 2" I W I.

Proof. Let B = B(eo; el, ..., e,,). For j - 1, 2, let

n

wJ -eo+Exijei E W c int(B)

and

bi - eo + giiei E vert(B).

i-1

Then -1 < xij < 1 and Aid E { 1, -1 } for i - 1, ... , n and j - 1, 2.

If

wl+bl

w2+b2

2

=

2 '

then

and so

for i = 1, ..., n. Since and

it follows that

II n

E(xiI + lt1 )ei - (xi2 + Ai2)ei,

i-I i-I

xil -X2=12i2-Ail

-2<xil-xi2<2

hit - hi l E (0, 2, -2),

xil-xi2=hi2-Ail=0

for i = 1, ... , n, and so wl = w2 and bl - b2. Therefore, I mid(W, vert(B)) I = I W I Ivert(B)I = 2" I W I.

Also, (x,;+Aij)/2 E (-1, 1) for i = 1, .

^.., n, and so (wj +b1)/2 E int(W); hence mid W, vert(B)) C int(B).

154 5. Sums of vectors in Euclidean space

Lemma 5.12 Let n > 2 and 1 < c < 2". Define so - eo(n, c) > 0 and k' -

k'(n, c) by

60- and

2" - c

4n(3"c+2nc+ 1)(4c)2"-'

k' - k*(n, c) -

^(4c)2-1.

If V is an n-dimensional vector space, and if A c V satisfies

JAI>k',

and

12A1 <-cIAJ,

IA n HI < soIAI

for every hyperplane H in V, then there exists W c A such that IWI > soIAI

and

Proof. We defined

12W1 <(c-so)IWI.

for any set W C V. and so Imid(W) l - 12W1. Let rw (u) denote the number of sets (WI - W2) C W such that (w, + w2)/2 - v. Then

1212 1W1(I21+1)

-

_rw(v)

vEmid(w)

< Imid(W)I max{rw(v) I v E mid(W)}.

Let A C V satisfy the three conditions of the lemma. We shall prove by induction that there exist hyperplanes H, , ... , H" in V, vectors fo, fl,..., E V with

fo -0, and sets A,__ A, such that

(i) f; EHjforO<i < j <n,

W * ' )

(iii) A. C A"_, C ... C A, C AO - A.

(iv) kj - IA,I >

₍₄₎

ki,_i for j - 1,...,n,

(v) Aj c (l;_, H,(+') for j - 1, ... , n, and

(vi) Aj+, U (12fj} - Aj+i) C Aj f o r j -0, 1,...,n - 1.

The proof is by induction. Let AO - A and ko - IA0I Choose fo E AO such that rA0(fo) - max{rA0(v) I V E mid(Ao)}.

Then

k02

2

and so

< r,4o(v)

vemid(Ao)

< rAo(fo)jmid(Ao)I

-

rAo(fo)I2AoI

< rAo(fo)Cko,

rAo(f0) > ko

Replacing A by A - { fo}, we can assume without loss of generality that fo - 0.

Let Hi be any hyperplane in V such that 0 - fo E HI. If (w, + w2)/2 - fo, then either { w, , W2) C H, or

I(w,, w2) n ^H,+')I

-

_{w,, w2) n} ^H(_,)I

_{- 1.}

Let

A,-{wEAoIwEH(+i)

and 2fo-wEAo}.

Then

A, c Ao n H(+') C H(+') and

Al U (12fo) - A,) S A.

Since IAo n H, I < eoko and

1 1

EO <

4c, it follows that

k, -IAAI

rA0(fo)-IAonH,I

> 2 - coko

ko 4c Thus, H,, fo, and A, satisfy conditions (i)-(vi).

156 5. Sums of vectors in Euclidean space

Suppose that I < m < n - 1, and that we have constructed hyperplanes

Hl,..., H,,,, vectors fo, fl, ..., fm _ 1, and sets A 1, .... A,,, that satisfy conditions (.)_(vi). Since A,,,

'c nm H(+I) and (l"'

_f

H(+I) is convex, it follows that

t ;_i _{i i-1 i}

m

mid(Am) C n

^H`+I)

i-

Choose E mid(Am) such that

I V E mid(Am)j.

Then

and so

2

((4C)2"_1)2

< 2

< rA_(fm)Imid(Am)I

5

^rA.(fm)I

rA., (fm )CkO,

2ko rA..(fm) > )2

Let Hm+1 be any hyperplane in V such that

{fo.fl,...,fm}C Hm+1.

Let

Then

and

Since

and

Am+l - 1W E Am I W E Hm+1 and 2 fm - W E Am }.

m+1

Am+1 C A. n Hm+1, C ^nH'(+1)

i-

Am+1 U (12f.) - Am+I) C Am.

IAmnHm+II 5 IAOn H.+11 5 E0ko

1

EO < (4c)2'+'- 1^,

it follows that

km+1

-

^{IAm+1 l}

2k0

(4c)2 "_1 ^EOkO

>

-

ko

Thus, the hyperplanes HI,..., H..,,, the vectors fo, fl,..., fm, and the sets

A,_., Am+1 satisfy conditions (i) -(vi) . This completes the induction.

Let a E An. Construct sets

S.(a)CSn-1(a)C...CSo(a)CA

by setting S. (a) - (a) and

Sj-1(a) - Sj (a) a ({2fj-1 } - Sj (a)) for j - 0, 1, ... , n - 1. We shall prove by induction that

m

Sm(a) C A. C n Hi(+

i-1

for m - 0, 1, ... , n. Clearly,

Suppose that

n

S.(a)-{a}C

^AnCnH(+1)

i-I

Sm+1(a) C Am+l C

m+I_Hi

+n

where 0 < m < n - 1. Then

and so

m

{2fm} - Sm+1(a) C {2fm} - Am+1 C Am Sn H(+1),

i-I

Sm(a) - Sm+1(a) U (12f.) - Sm+1(a)) C A. C n^H(+

i-

This completes the induction.

W e have shown that the hyperplanes HI, ... , Hn, the vectors

.fo, fl,..., f.-I, the set A. C H(1, ... , 1), and the sets

(Sk(a)IaEA. and k-0,1....,n}

satisfy the hypotheses of Theorem 5.3. Therefore, there exists a vector a* E An such that the block

B(a*) - conv(So(a`)) has the property that

IAnB(a')j > I U So(a) nB(a')

aEA.

IA,I

k

> ko(4c)2'_I.

158 5. Sums of vectors in Euclidean space

The block B(a') determines the 2n facial hyperplanes F1,,,,. where j - 1, ... , n

and tLj E It, -1j. Let

"

F' - U U

!-1

Let WO - A n int(B(a')). Because IA n HI < eoko for every hyperplane H and so < 4n(4C)21-1'1

it follows that and so

IAnF'I <2neoko,

I Wol

-

IA n int(B(a'))I

> IAnB(a')I-IAnF'I

ko - 2nsoko (4c)2"-1

ko

2(4c)2"-1.

Since

vert(B(a')) - So(a') a A,

it follows from Lemma 5.11 that

mid(Wo, So(a')) -{

^{w2 s} I W E Wo, S E So(a*))

c

int(B(a')) n mid(A)

and

Imid(Wo, So(a' )) I

-

2" I Wo I .

The 2n facial hyperplanes Fj.,,, partition V \ F' into 3" pairwise disjoint open convex sets where (A1.... , A,,) E (0, 1, -1)" and

Then

and

D(0... 0) - int(B(a')).

Wo-Anint(B(a'))-AnD(0,...,0)

mid(Wo. So(a*)) c int(B(a')) n mid(A) - D(0, ..., 0) n mid(A).

Let W1, ... , W3. -1 be the pairwise disjoint sets

A n D(A1,...,An),

where (11, ... , )") E 10, 1, -1)" and (A 1, ... , A,,) 71 (0, ... , 0). Since the sets D(,11... L") are convex, it follows that

mid(Wi) n mid(WW) - 0

forl <i < j<3"- 1, and

M(Wo. So(a*)) n mid(W;) - 0

fori-1, ,3"-1.

We shall prove that there exists i E 11, 2, ..., 3" - 1) such that the set W1 satisfies the conditions

IWiI?EOIAI - sko

and

12Wil - Imid(Wi)I < (c-so))Wjl

Suppose not. Then Imid(Wi)I > (c - Eo)IW11 for every set W1 satisfying 1 Wi I eoko. Let E' denote the sum over all i E [ 1, 3" - 1 ] such that I Wi I > Eoko. Then

cko _{>_ 12A1}

This implies that

and so

Imid(A)I

Y-I

> Imid(Wo, S(a`))I + EImid(Wj)I

i-I

> 2"IWoI+E Imid(Wi)I 2"IWoI+(c-so)E IWiI

> 2"IWoI +cE IWiI -soko

Y-I

> 2"IWoI+cEIW;I-3"csoko-EOko

i-I

3"-1

- (2" - c)IWol+cE I Wil -3"cEoko-Eoko

i-O

- ^{(2" -}

c)I Wo 1 + c(ko - I A n F* I) - 3"csoko - e0ko

?

(2" - c) I Wo I + cko - (3" c + 2nc + I )soko.

(3"c+2nc+ 1)EOko > (2" - c)IWoI > (2" - c)ko

2(4c)2^-1

2" - c

80

> 2(3"c+2nc+

^{1)(4c)2^-1} - 2nEo > Eo,

160 5. Sums of vectors in Euclidean space

which is absurd. Therefore, there exists a set W = W; e_ A such that I WI ? eoIAI and 12 W I < (c - e )l W 1. This completes the proof.

Proof of Theorem 5.1. For I < c < 2", let

eo - eo(n,c)

2" - c

4n(3"c+2nc+ 1)(4c)2"-l

be the positive real number defined in Lemma 5.12. Let t = t(n, c) be the unique positive integer such that

t-1 < c-I <t.

Let

eo

r

e0r = E0,

k' - k*(n, c) = ko = ko(n,c) = eo'k'.

eo(n, c) < eo(n, c') k*(n, c) > k*(n, c').

Let A be a subset of V such that

Al I> ko > ko and

12A1 < cIAl.

Suppose that

IA n HI <- eolAl 5 eolAl

for every hyperplane H in V. By Lemma 5.12, there exists a set W C- A such that

IWI?eoIAl?eoku=eo" i)k.>k*

and

12W1 <(c-Eo)IWI.

Moreover, for every hyperplane H in V,