Linearly independent hyperplanes

2 JAI,

5.2 Linearly independent hyperplanes

Let n > 2, and let V be an n-dimensional Euclidean space with inner product

( , ). Let h be a nonzero vector in V. and let y E R. Define H, H(+'), and H(-') as follows:

H - IV EV I (h,v)-y}

H(+')

-

IV E V I (h, v) > y)

H(-') - IV EV I (h,v)<y}.

The sets H(+') and H(-') are, respectively, the upper and lower open half-spaces determined by H. The vector h is called a normal vector to the hyperplane H. If 0 E H. then H is an (n - 1)-dimensional subspace of V. Note that 0 E H if and only if y - 0.

The set K in R" is convex if a, b E K implies that to + (1 - t)b E K for all t E [0, 1 ]. The sets H, H("), and H(-') are convex. For any subset Sofa Euclidean space V, the convex hull of S, denoted conv(S), is the smallest convex subset of V that contains S. Since the intersection of convex sets is convex, it follows that the convex hull of S is the intersection of all convex sets containing S. This intersection is nonempty since the Euclidean space V is convex and contains S.

Let H1, ... , H. be hyperplanes, and let H' - U 1 Hi. Let (1, -1 }m denote

the set of all m-tuples (µ 1 , ... , µm) such that µi E ( 1 , -1 ) f o r i -

1, ...

, m.For

(Iz1,...

E (1, -1 }m, let

-n (w

m

µm) I Hi

i-

The 2m sets H(µ1, ..., µm) are pairwise disjoint, and

V \ H' - U

H(Is1, .... Am).

(N... R.)EI1.-l I'

Let V be an n-dimensional Euclidean space, and let Hl , ... , H be hyperplanes with normal vectors h 1 , ... , hm, respectively. The hyperplanes H1, ... , H,,, are linearly independent if the vectors h1, ... , hm are linearly independent. The hy- perplanes Hl , ... , Hm are linearly dependent if the vectors h i , ... , h", are linearly dependent.

136 5. Sums of vectors in Euclidean space

Lemma 5.1 Let Hl , ... , Hbe hyperplanes in an n-dimensional Euclidean space V. Suppose that 0 E H; for all i - 1, ... , m. The hyperplanes Hl , ... , Hm are linearly independent if and only if

H(E.1.1,...,µm)710 for all (µ1, ..., µm) E {1, -I }m.

Proof. There exist nonzero vectors h1, ... ,in V such that

Hi-IVEVI(hi,v)-0}

for i - I, ... , m. Suppose that H1, ... , Hm are linearly independent hyperplanes.

Then the vectors h 1, ... , hm are independent, and there exists a dual set of vectors

h ,

^.. . , h,*, suchthat

(hr,hi)-s,. - _0ifi7(j ^{l if i - j}

fori, j - 1,...,m.Let(µ1,...,{Lm) E (1, -1)'. Let

m

v - Eµih! E V.

I-1 Then

and so

m

(hi, v)->µi(hi.hi)-tLi

i-1

V E 1

for all i - I, ... , m. It follows that N(µ1, ... µ) 710.

Suppose that H1, ... Hm are linearly dependent hyperplanes. Then the vectors h 1, ... , hm are dependent, and there exist scalars a1, ... , am not all zero such that

E_1a;h;-0. Define

+1

if ai > 0

µi

(

-1

if ai < 0.

Then µiai > 0 for all i - and µiai > 0 for some j.

We shall show that H(µ1, ... , µm) - 0. If not, choose v E H(µ1, ... , µm).

Then V E H,1"'1 implies that

ai(hi, v) > 0

for i - 1, ..., m, and

It follows that

ai(hi,v) > 0.

m

0- (0, v)- (aaht.v -Eai(hi,v)>0,

r-1 .-1

which is impossible. This proves that N(µ1, ... , µm) - 0.

Lemma 5.2 Let V be an n-dimensional Euclidean space, and let Hl,

..

Hr be linearly independent hyperplanes in V such that 0 E H , f o r i - 1, ... , r. Then

r

dim

(flH); -n-r.

In particular, if r - n, then

n

n Hi - (0).

Proof. Let h, , ... , hr be normal vectors for the hyperplanes Hl, ... H, respec- tively. Then the set of vectors (h, , ... , hr ) is linearly independent, and (hi, v) - 0 for all v E H;. Let w - f;_, Hi, and let

Wl -{vE V I(v,w)-0forallwE W}.

Since h; E W -L for i - l , ... , r, it follows that dim(W ) > r, and so

dim(W) - n - dim(W') < n - r.

We shall prove that dim(W) > n - r by induction on r. If r - 1, then W - H, and dim(W) - dim(HI) - n - 1. Let 2 < r < n, and assume that the assertion is

true for r - 1. Let W' -n

it Hi. Then dim(W') > n - r + 1. Since W - W' n Hr, it follows that

dim(W)

-

dim(W' n HH)

-

dim(W') + dim (Hr) - dim (W' + Hr)

> dim(W') + dim (Hr) - dim V

> (n-r+I)+(n-1)-n

=

n-r.

Therefore, dim(W)-n-r.Ifr-n, then dim(f =, H;)-0,andsof;_i Hi =(0}.

Lemma 5.3 Let V be an n-dimensional Euclidean space and let HI .., H,,, be hyperplanes in V such that 0 E H; for i - 1, ..., m. Let S be a subset of V such that

SnH(µ,,...,k,,,)-/ 0

(5.1)

forall(tl,,...,µ,,,)E (1,-1)m.Then

conv(S) n ; ,'0.

Proof. By induction on m. Let m - 1, and let h, be a normal vector for the hyperplane Hl. By (5.1), there exist vectors s, E S n ^Hi+,, and S2 E S n HI-¹ such that

(h1,s,)-oil > 0

138 5. Sums of vectors in Euclidean space and

(h i, s2) - -

Then

S - 1 I s) +

(

a) +a2

I S2 E conv(S).

Since

- ,a2 - (11 C12 (h" S) - - 0,

a2(hI,s))+ac(h1,s2) a

a ,+a2 a,+a2 a ,+a2

it follows that $ E HI, and so

conv(S)fl H) 1 0.

Let m > 2, and suppose that the lemma holds for m - 1. Define S(+)) and S(-)) by

S

n

H,(.- Then (5.1) implies that

Then

S(+))n (n Hcµ,)m-i

-SnH(µ),...,IA.-,.+1)710

r-,

and m-I

s(- 1) n H(N,)) S f H(FLI, ... , lam-c, -1)10

f o r all (JAI,

km-1) E (1, -1

^}m-). By the induction hypothesis, the lemma holds for the m - I hyperplanes H,, ... , Hm-1, and so

m-I

cony (S(+')) n n H;) 10.

This means that there exist vectors s) , ... , sk E S(+') and scalars a1,... ak E R

with a; > 0 for i - I... k and a, + ... + ak - I such that

k m-1 -1

s(+I) - E a; s; E conv(S('))) n n H; c_ con(s) n

m

H;

Moreover, s(+) E H.(+') since Si E

for i - I... k and

is convex.

Similarly, there existss

a2 < 0.

m-) n,-I

s(-1) E cony (s(-1)) n n H;

con(s) n (n

^{H; I ,}

ands(-') E

Let T - {s(''), s(-0). Then

m-I 1

T c con(s) n n Hit ,

and so

m-i

conv(T) c con(s) n n Hi

^.

i-i

Since T n f 0 for µm E { 1, -1 }, it follows that there exists

s c- conv(T) n H. c con(s) n (Hi).

This completes the proof.

Lemma 5.4 Let V be an n-dimensional Euclidean space and let H1, ..., H be hyperplanes in V such that 0 E H, for all i - 1, ... , n. Let S be a subset of V such that

SnH(µi,...,µn)-SnlnH,ci)

// ⁰

i- for all

µn) E 11, -1)". Then

0 E conv(S).

Proof. It follows from Lemma 5.1 that the hyperplanes H1, ..., H,, are linearly independent and so, by Lemmas 5.2 and 5.3,

n

conv(S) n {0} - conv(S) n

(n Hil

0.

f ) /

Lemma 5.5 Let V be an n-dimensional vector space, and let H1... H be lin- early independent hyperplanes in V with normal vectors h 1 , ... , h,,, respectively, and with 0 E Hi for all i - 1, ... , n. Let

n

Lj -nHi

;ri

forj-l....,n.Then

V - Hj®Lj

for j - 1, ... , n. Moreover, there exists a dual basis {h,, ... , h,, } for V such that

I

ifi-j

(hi, hi) - ai.j ⁰

if i 7Q

and Hj is the (n - 1)-dimensional subspace spanned by h,,...,hj_t, h!+,, ...

,

hn and L j is the one-dimensional subspace spanned by h*.

140 S. Sums of vectors in Euclidean space

Proof. It follows from Lemma 5.2 that dim(Lj) = 1. Let fj* be a basis vector

for L. Then fi* E Lj implies that ff E H, for all i ¢ j, and so (hi, f'.) = 0 for

i j. Moreover,

(hj,fj)=0

if and only if

fj* EHjnLj=nH,=(0},

i_1

which is impossible since fJ 0. Therefore, (hj, 0 and

h' fi

ELj\Hj.

(h,, f;*)

Then (hi, h*) = Si,j for i, j = 1, ..., n, and Lj is spanned by h*. The vectors /t" are linearly independent, since _, xjh* = 0 implies that

0=(h,,0)=(hi,xjhf)xj(hi,h*)=xi

It

j-1 l-1

for i = 1, ..., n. Moreover, (hi, h*) = 0 for all i j implies that h, E Hj for i f j. Since Hj is a vector subspace of dimension n - 1, it follows that the set {hi,...,h*_i,h,... h) is a basis for H,, and V H

L. This completes the proof.

Lemma 5.6 Let V be an n -dimensional vector space, and let H1, ... , H be lin- early independent hyperplanes in V with 0 E H i f o r i = 1, ... , n. Let Qi = Hi n H,, f o r i = 1 , ... , n - 1 . Then dim Qi = n - 2, and Q1 ,

pendent hyperplanes in H,,. Let

Q are linearly inde-

-i

L,, =nH1,

i_1

and let 7r : V -* H be the projection corresponding to the direct sum decompo- sition

V = H,, ® L,.

Let (tt 1, ... , E (1, - I and let V E V. If

,1 1

uEnHi')

then

,r(u) E n Qlu,)

i_i

Let S be a subset of V, and let rr(S) - (7r (s) S E S} c H,,. If

Sn H,")

⁰

for all (µ,. ....

E (1, -l, then 0 E Conv(ir(S)).

There exists a basis vector h,* for L,, such that if S C then ah, E Conv(S)

for some a > 0.

Proof. It follows from Lemma 5.2 that dim(Q;) - n - 2, and so Qi is a hyper-

plane in H for i - 1, ... , n - 1.

Let h, , ... , h be normal vectors f o r H, , ... , H,,, respectively, and let (h,, , h; )

qi

-

^h,

^(h,

Since 1:1, ... , h,, are linearly independent vectors, it follows that the vectors

q,, ... ,

are linearly independent. Moreover, (h,,, qi) - 0, and so qi E H,,

fori-1,...,n-1.

Letw E H,,.Then (gi,w)-(hi,w),and so w E Qi if and only if(gi,w)-0.

Thus, qi is a normal vector for Qi in the vector subspace H,,, and the hyperplanes

Q,, ... ,

are independent in H,,.

By Lemma 5.5, there is a basis vector h, for L,, such that (hi, h,) - 8i,,,. If v E V, then

v - 7r (v) +rp(v)hn*, where n(v) E H and ip(v) E R. Moreover,

(qi, 7r (v))

- (hi,'r(v)) - (h, h,,) (ha, r(v))

_ (hi, 7r(v))

(hi, v) - rp(v)(h;, h,*) (hi, v)

for i - 1, ... , n - 1. It follows that

,1-I

VEnN(`')

i-i

if and only if

7r (V) E Q0.,).

ii

142 5. Sums of vectors in Euclidean space

If

-i

Sn

n

nHw,>

710 i-1

for all (µ i , E ( 1, -1)"-', then

n-1

7r (s) n

(

^QcUO

) _{f g}

J

g

for a11(µ E (1, Lemma 5.4 implies that

0 E convOr(S)).

This means that there exist vectors sl, ... , sk E S with the property that

f o r some nonnegative scalars a,, ... , ak such that a, +... + ak - 1. Let si -7r(si)+(p(si)hn

and

Then

a - E amsi )- i-

k k k

aisi - a;7r(s;)+a;p(s;)h, -ahn E conv(S).

If S c

then

(h s;) - (h,, , 7r (s;)) + rp(s;)(h

h,) - rp(s,) > 0

for all i - 1, ..., k, and so a > 0 and ahn E conv(S). This completes the proof.