hyperplane separation

hyperplane separation

Let

X

be a

vector space

, and

Φ

be any

subspace

of

linear functionals

on

X

.
Impose on

X

the

weak topology

generated by

Φ

.

Theorem 1

(Hyperplane Separation Theorem I).

Given a weakly closed convex subset

S

⊂

X

,
and

a

∈

X

∖

S

.
there is

ϕ

∈

Φ

such that

ϕ

⁢

(

a

)

<

inf

x

∈

S

⁡

ϕ

⁢

(

x

)

.

Proof.

The weak topology on

X

can be generated by the

semi-norms

x

↦

|

p

⁢

(

x

)

|

for

p

∈

Φ

.
A subbasis for the weak topology
consists of neigborhoods of the form

{

x

∈

X

:

|

p

⁢

(

x

-

y

)

|

<

ϵ

}

for

y

∈

X

,

p

∈

Φ

and

ϵ

>

0

.
Since

X

∖

S

is weakly open,
there exist

f

1

,

…

,

f

n

∈

Φ

and

ϵ

>

0

such that

|

f

i

⁢

(

x

)

-

f

i

⁢

(

a

)

|

=

|

f

i

⁢

(

x

-

a

)

|

<

ϵ

,

for all

i

=

1

,

…

,

n

implies

⁢

x

∈

X

∖

S

.

In other words, if

x

∈

S

then at least one of

|

f

i

⁢

(

x

)

-

f

i

⁢

(

a

)

|

is

≥

ϵ

.

Define a map

F

:

X

→

ℝ

n

by

F

⁢

(

x

)

=

(

f

1

⁢

(

x

)

,

…

,

f

n

⁢

(

x

)

)

.
The set

F

⁢

(

S

)

¯

is evidently
closed and convex in

ℝ

n

, a

Hilbert space

under the standard

inner product

.
So there is a point

b

∈

F

⁢

(

S

)

¯

that minimizes the norm

∥

b

-

F

⁢

(

a

)

∥

.

It follows that

⟨

y

-

b

,

b

-

F

⁢

(

a

)

⟩

≥

0

for all

y

∈

F

⁢

(

S

)

¯

;
for otherwise we can attain a smaller value of the norm
by moving from the point

b

along a line towards

y

.
(Formally, we have

0

≤

d

d

⁢

t

|

t

=

0

⁢

∥

t

⁢

y

+

(

1

-

t

)

⁢

b

-

F

⁢

(

a

)

∥

2

=

2

⁢

⟨

y

-

b

,

b

-

F

⁢

(

a

)

⟩

.)

Take

ϕ

=

∑

i

=

1

n

λ

i

⁢

f

i

where

λ

=

b

-

F

⁢

(

a

)

.
Then we find, for all

x

∈

S

,

ϕ

⁢

(

x

-

a

)

=

⟨

b

-

F

⁢

(

a

)

,

F

⁢

(

x

-

a

)

⟩

=

⟨

b

-

F

⁢

(

a

)

,

b

-

F

⁢

(

a

)

⟩

+

⟨

b

-

F

⁢

(

a

)

,

y

-

b

⟩

,

y

=

F

⁢

(

x

)

∈

F

⁢

(

S

)

¯

≥

∥

b

-

F

⁢

(

a

)

∥

2

+

0

≥

ϵ

2

.

∎

Theorem 2

(Hyperplane Separation Theorem II).

Let

S

⊂

X

be a weakly closed convex subset,
and

K

⊂

X

a

compact

convex subset,
that do not

intersect

each other.
Then there exists

ϕ

∈

Φ

such that

sup

y

∈

K

⁡

ϕ

⁢

(

y

)

<

inf

x

∈

S

⁡

ϕ

⁢

(

x

)

.

Proof.

We show that

S

-

K

=

{

x

-

y

:

x

∈

S

,

y

∈

K

}

is weakly closed in

X

.
Let

{

z

α

=

x

α

-

y

α

}

⊆

A

be a net convergent to

z

.
Since

K

is compact,

{

y

α

}

has a subnet

{

y

α

⁢

(

β

)

}

convergent to

y

∈

K

.
Then the subnet

x

α

⁢

(

β

)

=

z

α

⁢

(

β

)

+

y

α

⁢

(

β

)

is convergent to

x

=

z

+

y

.
The point

x

is in

S

since

S

is closed;
therefore

z

=

x

-

y

is in

S

-

K

.

Also,

S

-

K

is convex since

S

and

K

are.
Noting that

0

∉

S

-

K

(otherwise

S

and

K

would have a common point),
we apply the previous

theorem

to
obtain a

ϕ

∈

Φ

such
that

0

=

ϕ

⁢

(

0

)

<

inf

z

∈

S

-

K

⁡

ϕ

⁢

(

z

)

≤

ϕ

⁢

(

x

-

y

)

,

for all

x

∈

S

and

y

∈

K

.

The desired

conclusion

follows at once.
∎

Title

hyperplane separation

Canonical name

HyperplaneSeparation

Date of creation

2013-03-22 17:19:01

Last modified on

2013-03-22 17:19:01

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

4

Author

stevecheng (10074)

Entry type

Theorem

Classification

msc 46A55

Classification

msc 49J27

Classification

msc 46A20

Synonym

separating hyperplane

Related topic

HahnBanachgeometricFormTheorem

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