Sperner’s lemma

Sperner’s lemma

Let

A

⁢

B

⁢

C

be a

triangle

, and let

S

be the set of vertices of some

triangulation

T

of

A

⁢

B

⁢

C

. Let

f

be a

mapping

of

S

into a three-element set, say

{

1

,

2

,

3

}

=

T

(indicated by
red/green/blue respectively in the figure), such that:

•

any point

P

of

S

, if it is on the side

A

⁢

B

,

satisfies

f

⁢

(

P

)

∈

{

1

,

2

}

•

similarly if

P

is on the side

B

⁢

C

, then

f

⁢

(

P

)

∈

{

2

,

3

}

•

if

P

is on the side

C

⁢

A

, then

f

⁢

(

P

)

∈

{

3

,

1

}

(It follows that

f

⁢

(

A

)

=

1

,

f

⁢

(

B

)

=

2

,

f

⁢

(

C

)

=

3

.)
Then some (triangular) simplex of

T

, say

U

⁢

V

⁢

W

, satisfies

f

⁢

(

U

)

=

1

f

⁢

(

V

)

=

2

f

⁢

(

W

)

=

3

.

We will informally sketch a proof of a

stronger

statement:
Let

M

(resp.

N

) be the number of simplexes satisfying (1) and whose
vertices have the same

orientation

as

A

⁢

B

⁢

C

(resp. the
opposite orientation). Then

M

-

N

=

1

(whence

M

>

0

).

The proof is in the style of well-known proofs of, for example, Stokes’s

theorem

in the plane, or Cauchy’s theorems about a

holomorphic function

.

Define an

antisymmetric

function

d

:

T

×

T

→

ℤ

by

d

⁢

(

1

,

1

)

=

d

⁢

(

2

,

2

)

=

d

⁢

(

3

,

3

)

=

0

d

⁢

(

1

,

2

)

=

d

⁢

(

2

,

3

)

=

d

⁢

(

3

,

1

)

=

1

d

⁢

(

2

,

1

)

=

d

⁢

(

3

,

2

)

=

d

⁢

(

1

,

3

)

=

-

1

.

Let’s define a “circuit” of size

n

as an

injective

mapping

z

of the
cyclic group

ℤ

/

n

⁢

ℤ

into

V

such that

z

⁢

(

n

)

is adjacent to

z

⁢

(

n

+

1

)

for all

n

in the group.

Any circuit

z

has what we will call a

contour integral

I

⁢

z

, namely

I

⁢

z

=

∑

n

d

⁢

(

z

⁢

(

n

)

,

z

⁢

(

n

+

1

)

)

.

Let us say that two vertices

P

and

Q

are

equivalent

if

f

⁢

(

P

)

=

f

⁢

(

Q

)

.

There are four steps:

1) Contour integrals are added when their corresponding circuits are juxtaposed.

2) A circuit of size 3, hitting the vertices of a simplex

P

⁢

Q

⁢

R

, has contour integral

•

0 if any two of

P

,

Q

,

R

are equivalent, else

•

+3 if they are inequivalent and have the same orientation as

A

⁢

B

⁢

C

, else

•

-3

3) If

y

is a circuit which travels around the perimeter of the whole
triangle

A

⁢

B

⁢

C

, and with same orientation as

A

⁢

B

⁢

C

, then

I

⁢

y

=

3

.

4) Combining the above results, we get

3

=

∑

I

⁢

w

=

3

⁢

M

-

3

⁢

N

where the sum contains one summand for each simplex

P

⁢

Q

⁢

R

.

Remarks:

In the figure,

M

=

2

and

N

=

1

: there are two “red-green-blue” simplexes and one blue-green-red.

With the same hypotheses as in Sperner’s lemma, there is such a simplex

U

⁢

V

⁢

W

which is

connected

(along edges of the triangulation) to
the side

A

⁢

B

(resp.

B

⁢

C

,

C

⁢

A

)
by a set of vertices

v

for which

f

⁢

(

v

)

∈

{

1

,

2

}

(resp.

{

2

,

3

}

,

{

3

,

1

}

)

. The figure illustrates that result: one of the red-green-blue
simplexes is connected to the red-green side by a red-green “curve”,
and to the other two sides likewise.

The original use of Sperner’s lemma was in a proof
of Brouwer’s fixed point theorem in two

dimensions

.

Title

Sperner’s lemma

Canonical name

SpernersLemma

Date of creation

2013-03-22 13:44:33

Last modified on

2013-03-22 13:44:33

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

12

Author

mathcam (2727)

Entry type

Theorem

Classification

msc 55M20

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