KKM lemma

KKM lemma

1

Preliminaries

We start by introducing some standard notation.

ℝ

n

+

1

is the

(

n

+

1

)

-dimensional real space with

Euclidean norm

and
metric. For a subset

A

⊂

ℝ

n

+

1

we denote by

diam

⁡

(

A

)

the

diameter

of

A

.

The

n

-dimensional

simplex

𝒮

n

is the following
subset of

ℝ

n

+

1

{

(

α

1

,

α

2

,

…

,

α

n

+

1

)

|

∑

i

=

1

n

+

1

α

i

=

1

,

α

i

≥

0

∀

i

=

1

,

…

,

n

+

1

}

More generally, if

V

=

{

v

1

,

v

2

,

…

,

v

k

}

is a set of vectors,
then

S

⁢

(

V

)

is the simplex

spanned by

V

:

S

(

V

)

=

{

∑

i

=

1

k

α

i

v

i

,

|

∑

i

=

1

k

α

i

=

1

,

α

i

≥

0

∀

i

=

1

,

…

,

k

}

Let

ℰ

=

{

e

1

,

e

1

,

…

,

e

n

+

1

}

be the standard

orthonormal basis

of

ℝ

n

+

1

. So,

𝒮

n

is
the simplex spanned by

ℰ

. Any

element

v

of

S

⁢

(

V

)

is
represented by a vector of

coordinates

(

α

1

,

α

2

,

…

,

α

k

)

such that

v

=

∑

i

α

i

⁢

v

i

; these are called a

barycentric coordinates

of

v

. If the
set

V

is in

general position

then the above

representation

is
unique and we say that

V

is a basis for

S

⁢

(

V

)

. If we write

S

⁢

(

V

)

then

V

is always a basis.

Let

v

be in

S

⁢

(

V

)

,

V

=

{

v

1

,

v

2

,

…

,

v

k

}

a basis and

v

represented (uniquely) by barycentric coordinates

(

α

1

,

α

2

,

…

,

α

k

)

. We denote by

F

V

⁢

(

v

)

the
subset

{

j

|

α

j

≠

0

}

of

{

1

,

2

,

…

,

k

}

(i.e., the set of non-null coordinates). Let

I

⊂

{

1

,

2

,

…

,

k

}

, the

I

-th face of

S

⁢

(

V

)

is the set

{

v

∈

S

⁢

(

V

)

|

F

V

⁢

(

v

)

⊆

I

}

. A

face

of

S

⁢

(

V

)

is an

I

-face for some

I

(note that this is

independent

of the choice
of basis).

2

KKM Lemma

The main result we prove is the following:

Theorem 1

(Knaster-Kuratowski-Mazurkiewicz Lemma

[

1

]

).

Let

S

n

be the standard simplex spanned by

E

the standard orthonormal basis for

R

n

+

1

. Assume we have

n

+

1

closed subsets

C

1

,

…

,

C

n

+

1

of

S

n

with the property that for
every subset

I

of

{

1

,

2

,

…

,

n

+

1

}

the following holds: the

I

-th face of

S

n

is a subset of

∪

i

∈

I

C

i

.
Then, the

intersection

of the sets

C

1

,

C

2

,

…

,

C

n

+

1

is
non-empty.

We prove the KKM Lemma by using Sperner’s Lemma; Sperner’s Lemma
is based on the notion of simplicial

subdivision

and

coloring

.

Definition 2

(Simplicial subdivision of

S

n

).

A

simplicial subdivision

of

S

n

is a couple

D

=

(

V

,

B

)

;

V

are the vertices, a finite subset of

S

n

;

B

is a set of simplexes

S

⁢

(

V

1

)

,

S

⁢

(

V

2

)

,

…

,

S

⁢

(

V

k

)

where each

V

i

is a subset of

V

of
size

n

+

1

.

D

has the following properties:

1.

The union of the simplexes in

ℬ

is

𝒮

n

.

2.

If

S

⁢

(

V

i

)

and

S

⁢

(

V

j

)

intersect then the
intersection is a face of both

S

⁢

(

V

i

)

and

S

⁢

(

V

j

)

.

The

norm

of

D

, denoted by

|

D

|

, is the diameter of the
largest simplex in

B

.

An

(

n

+

1

)

-

coloring

of a subdivision

D

=

(

V

,

ℬ

)

of

𝒮

n

is a

function

C

:

V

→

{

1

,

2

,

…

,

n

+

1

}

A

Sperner Coloring

of

D

is an

(

n

+

1

)

-coloring

C

such
that

C

⁢

(

v

)

∈

F

ℰ

⁢

(

v

)

for every

v

∈

V

, that is, if

v

is on the

I

-th face then its color is from

I

. For example,
if

D

=

(

V

,

ℬ

)

is a subdivision of the standard simplex

𝒮

n

then the

standard basis

ℰ

is a subset
of

V

and

F

ℰ

⁢

(

e

i

)

=

i

. Hence, If

C

is a Sperner
Coloring of

D

then

C

⁢

(

e

i

)

=

i

for all

i

=

1

,

2

,

…

,

n

+

1

.

Theorem 3

(Sperner’s Lemma).

Let

D

=

(

V

,

B

)

be a simplicial subdivision of

S

n

and

C

:

V

→

{

1

,

2

,

…

,

n

+

1

}

a Sperner Coloring
of

D

. Then, there is a simplex

S

⁢

(

V

i

)

∈

B

such that

C

⁢

(

V

i

)

=

{

1

,

2

,

…

,

n

+

1

}

.

It is a standard result, for example by barycentric subdivisions,
that

𝒮

n

has a

sequence

of simplicial subdivisions

D

1

,

D

2

,

…

such that

|

D

i

|

→

0

. We use this fact to prove
the KKM Lemma:

Proof of KKM Lemma.

Let

C

1

,

C

2

,

…

,

C

n

+

1

be closed subsets of

𝒮

n

as given in the lemma. We define the following function

γ

:

𝒮

n

→

{

1

,

2

,

…

,

n

+

1

}

as follows:

γ

⁢

(

v

)

=

min

⁡

{

i

|

i

∈

F

ℰ

⁢

(

v

)

⁢

and

⁢

v

∈

C

i

}

γ

is well defined by the

hypothesis

of the lemma and

γ

⁢

(

v

)

∈

F

ℰ

⁢

(

v

)

. Also, if

γ

⁢

(

v

)

=

i

then

v

∈

C

i

. Let

D

1

,

D

2

,

…

be a sequence of simplicial
subdivisions such that

|

D

i

|

→

0

. We set the color of every
vertex

v

in

D

i

to be

γ

⁢

(

v

)

. This is a Sperner Coloring
since if

v

is in

I

-fact then

γ

⁢

(

v

)

∈

F

ℰ

⁢

(

v

)

⊆

I

. Therefore, by Sperner’s Lemma we have in each
subdivision

D

i

a simplex

S

⁢

(

V

i

)

such that

γ

⁢

(

V

i

)

=

{

1

,

2

,

…

,

n

+

1

}

. Moreover,

diam

⁡

(

S

⁢

(

V

i

)

)

→

0

.
By the properties of

γ

for every

i

=

1

,

2

,

…

and every

j

∈

{

1

,

2

,

…

,

n

+

1

}

we have that

S

⁢

(

V

i

)

∩

C

j

≠

ϕ

.
Let

u

i

be the arithmetic mean of the elements of

V

i

(this is
an element of

S

⁢

(

V

i

)

and thus an element of

𝒮

n

).
Since

𝒮

n

is bounded and closed we get that

u

i

has
a converging subsequence with a limit

L

∈

𝒮

n

. Now,
every set

C

j

is closed, and for every

ϵ

>

0

we have an
element of

C

j

of

ϵ

-

distance

from

L

; thus

L

is in

C

j

.

Therefore,

L

is in the intersection of all the sets

C

1

,

C

2

,

…

,

C

n

+

1

, and that proves the assertion.
∎

References

1

B. Knaster, C. Kuratowski, and S. Mazurkiewicz, Ein Beweis des
Fixpunktsatzes für n-dimensionale Simplexe,

Fund. Math. 14 (1929) 132-137.

Title

KKM lemma

Canonical name

KKMLemma

Date of creation

2013-03-22 16:26:31

Last modified on

2013-03-22 16:26:31

Owner

uriw (288)

Last modified by

uriw (288)

Numerical id

14

Author

uriw (288)

Entry type

Theorem

Classification

msc 54H25

Classification

msc 47H10

Synonym

K-K-M lemma

Related topic

BrouwerFixedPointTheorem

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