>.
0.
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ON KUHN'S STRONG CUBICAL LEMMA
by
Robert M. Freund
Sloan W.P. No. 1557-84
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
May, 1984
ON KUHN'S STRONG CUBICAL LEMMA
by
Robert M. Freund
Sloan W.P. No. 1557-84
May, 1984
Abstract
The Strong Cubical Sperner lemma of H.W. Kuhn asserts, under appropriate
conditions, the existence on the n-cube of an n-simplex with the label set
{0,..., n)
.
This lemma contrasts other results on the cube which assert the
existence of a 1-simplex with certain labels.
In this paper, we present a
Strong Simplotope theorem, which implies the Strong Cubical Sperner lemma and
Sperner'
s
lemma on the simplex.
This theorem is used to prove a set covering
theorem on the simplotope and on the cube.
We also report the relationship
between these results and Brouwer's fixed-point theorem.
1.
Introduction
In his 1960 article "Some Combinatorial Lemmas in Topology,"
[
7
]
H.W.
Kuhn presents his Strong Cubical Sperner lemma and its byproduct, the Cubical
Sperner lemma, where the appelation "Sperner" in the title is a reference to
the similarity of these lemmas with Sperner'
s
lemma on the simplex [11].
The development of fixed-point computational methods since 1967 has led to
renewed interest in Sperner'
s
lemma as well as other combinatorial analogs of
Brouwer's fixed-point theorem (see, for example, Scarf [9
Shapley [10], and more recently Fan
[
1 ],
and Freund
[
],
2 ],
[
Kuhn [8],
4 ],
[
5 ]).
Although many of these later results have built on ideas imbedded in Kuhn's
1960 paper (simplicial subdivision, pricewise-linear maps, equivalence with
Brouwer's theorem), Kuhn's Strong Cubical lemma has received little attention.
Some of the reason for this lack of attention is that the Strong
Cubical lemma differs structurally from other combinatorial results on the
simplex and cube.
Kuhn's lemma is based on a vector labelling of zeros and
ones, as opposed to an integer labelling in the case of Sperner' s lemma or the
cubical lemmas presented in [4].
Furthermore, it condenses the vector
labelling into a "reduced labelling" based on the number of initial zeroes in
the vector label, on ordering convention that is not used in the other com-
binatorial results in the fixed-point literature.
Finally, Kuhn's lemma
asserts the existence of the n-cube of an n-simplex with certain labels, i.e.
the dimension of the simplex under question equals the dimension of cube.
This latter "full-dimensionality" property is in contrast to the cubical
results shown in [5], which assert the existence of a 1-simplex on the
n-cube with certain complementary labels.
The purpose of this paper is to extend Kuhn's Cubical Sperner lemma
to the simplotope (which is the cross-product of simplices)
-1.0-
,
and in so doing
to provide some insight and perspective into the Strong Cubical Sperner lemma.
We prove a Strong Simplotope theorem, which implies the Cubical Sperner lemma
on the n-cube and the Sperner lemma on the simplex.
We also prove a Simplotope
Set Covering theorem, which implies the Knaster-Kuratowski-Mazurkiewicz
covering lemma on the simplex
[
6 ].
Finally, we show the equivalence of the
Strong Simplotope theorem with Brouwer's fixed-point theorem, and derive
Brouwer's theorem from the Simplotope Covering theorem.
The notation used herein is presented in Section 2.
Section
3
a condensation of the theory of labelled V-complexes, as presented in
This material is central to the proofs of the material which follows.
Section
4,
contains
[
3 ].
In
the Strong Simplotope theorem and the Simplotope Covering theorem
are stated and proved, and the relationship of these results to Brouwer's
theorem is demonstrated.
Section
5
presents the application of these results
to the cube and the simplex, and Section 6 contains some concluding remarks.
-1.1-
.
2.
Notation
Let
denote real n-dimensional space, and define e to be the vector of
IR
l's, namely e = (1,...,
cardinality of a set
m
Let v
,
.
.
.
denote the empty set, and let
<|>
For two sets
S.
be vectors in
v
,
Let
1).
IR
^
...
1
u
then the convex hull of v.0
,
j
_j <v uo , ...,
denoted
„m
.
.
.
v'",
,
v"'>
said to be a real m -dimensional simplex , or more simply an m-simplex.
(v ,..., v ) is an m-simplex and
a =
{v
.
,
.
.
v
,
}
then
,
(v
t =
,
.
.
.
T>.
m
1
1),
x
If the matrix
.
.0
has rank (m +
denote the
s|
S\T={x|xeS,
let
T,
S,
|
,
v
{v 0,..., v
>
is
If
is a nonempty subset of
}
is a k -face or face of a
Let H be an m-dimensional convex set in
IR
Let C be a collection
.
C is a triangulation
of m-simplices o together with all of their faces.
of H if
H =
i)
U
a
,
aeC
a
ii)
,
t
e
C imply a n t e C,
and
If o is an (m-1) -simplex of C, a is a face of at most two m-simplices
iii)
of C.
be locally finite if for each vertex v
C is said to
o e C that contain
v is a finite set.
S.,..., S
are n simplices in
R
m.
R,%
If
set
S
= S,
x
x S
n
in
IR
x
,
x
e
mn
m1
,
.
.
.
,
TR
,
H,
respectively, the
•
is called a simplotope
simplotope is the cross product of n simplices, for n >
simplex is itself a simplotope (by setting n = 1)
{x e IR
{x.
< x < e}
1
e
IR
|
the set of simplices
,
1.
.
Note that any
and the n-cube
is just the cross product of the n 1-simplices
<x. < 1},
j
= 1,..., n.
-2.0-
Thus a
.
Review of V-Complex Terminology and Results
3.
This section presents a condensation of the terminology and major
results concerning the theory of V-complexes, as presented in [3]
This material is central to proof
of the combinatorial theorem
An abstract complex consists of a set of vertices K
nonempty subsets of K
v
i)
ii)
<}>
e
^ x c y
e
e
K.
An element x of K is called in abstract simplex
If x e K and
cardinality.
= n + 1,
|x|
complex by K alone.
then x is called an n-simplex, where
|*|
denotes
K for which v
e
,
is implied by K, it is convenient to denote the
An abstract complex K is said to be finite if K
finite, and is locally finite if for each v € K
e
or more simply a simplex.
,
Technically, an abstract complex is defined by the pair (K
However, since the set K
x
and a set of finite
K
K implies x
e
in Section 4.
denoted K, such that
,
implies {v}
K
.
is
the set of simplices
,
x is a finite set.
An n-dimensional pseudomanifold , or more simply an n-pseudomanifold,
where n >
i)
ii)
x
1,
is a complex K such that
e
K implies there exists y
If x
e
K and
|x|
= n,
e
K with
|y|
= n
+
1
and x c y.
then there are at most two n-simplices of K
that contain x.
Let K be an n-pseudomanifold, where n > 1.
3K, is defined to be the set of simplices x
an (n -
1)
-simplex y
e
K,
The boundary of K, denoted
eK such that x
is contained in
and y is a subset of exactly one n-simplex of K.
A O-dimensional pseudomanifold K is defined to be a set of one of the
following two forms:
-3.0-
K)
i)
K =
{<(>,
{v}},
ii)
K =
U,
{u},
Because K contains
where K° = {v}, or
{v}}, where K° = {u, v}.
the empty set, as a member, K is not properly a complex
<£,
by the usual definition.
Here, however,
K is of type (i) above, we denote 3K =
9K =
<J>,
is defined as a -1-simplex.
$
If
If K is of type (ii) above, then
{<t>}.
i.e. K has no boundary.
If C is a triangulation of a set H in
IR
with vertex set K
,
then
corresponding to each simplex a in C is its set of vertices {v ,..., v
}.
Let K be collection of these sets of vertices together with their nonempty
subsets.
Then K is a pseudomanifold and K is defined to be the pseudomanifold
corresponding to
C.
Let K be a locally-finite abstract complex with vertices K
be a fixed finite nonempty set, called the label set
.
Let 7 denote a
.
collection of subsets of N, denoted the admissible subsets of N.
be a map A(*)
set S.
K, N,
:
FT
J
K
-*
2
\
<J>,
where
S
Let N
Let A(*)
denotes the collection of subsets of a
2
J, A(*) are said to constitute a V-complex with operator A(«)
and admissible sets 7, if the following eight conditions are met:
i)
K is a locally finite complex with vertices K
N
ii)
7c
2
iii)
T €
7
iv)
v)
vi)
vii)
viii)
A(-)
€
7
7+
2
S
,
:
For any x
If T e
e
7
7
\
K,
e
,
S
n
T
e
1
K
For any S, T
For T
implies
e
A(T)
and T
<j>
there is a T
such that x
j
e
J, A(S n T) = A(S)
n
e
A(T)
A(T)
is a pseudomanifold of dimension
u {j}
€
7
but
j
-3.1-
I
T,
|t|
then A(T) c 3A(T
u
{j}).
.
The nomenclature "V-complex" is short for variable-dimension complex,
and derives from property (vii) above, where the dimension of the
pseudomanifolds A(T) varies over the range of T
V-complex, for each x
If K is a
T
=
X
e
7.
K, we define:
e
T
n
Te7
xeA(T)
x is a full simplex if
as 3*A(T) = {x
9'A(<j>)
=
8A(T)
e
If
{$}.
=
|x|
T
|
=
A(<J>)
x
|t
= T}
+1.
j
.
If
For each T
<}>
7
e
and
{u}, {v}}, then
{<{>,
A(<(>)
9'A(<|>)
Let K be a V-complex with label set N.
e7, we also define
=
U,
3'A(T)
{v}}, then
= $
A function L(»)
:
K
+ N that
assigns an element of N to each vertex of K is said to be a labelling
function
L(x) =
u
is a labelling function,
If L(-)
.
L(v)
for each x
Two distinct simplices x, y
.
e
e K,
we define
K are defined to be adjacent
,
vex
written x ~
i)
ii)
if
y,
x and y are full, and
L(x
= T
n y)
J
x
u
T
.
y
Note that if x ~ y for some y, L(x) = T
"
x ~ y, L(x) 3
L(x
ny)=T x UT y
J
To see this, observe that if
.
=>T.
x
For a given V-complex K and labelling function L(«), we define the
two sets:
G = {x e K
|
x is full and L(x) =
K
|
x
B = {x e
e
3'A(T
)
T
,
and L(x) = T
G and B are short for "good" and "bad",
and L(x) ^7>, and
}.
for in most applications of V-complexes,
a path-following scheme will terminate with an element of G or B.
G typically
contains those simplices with pre-specif ied desirable properties, whereas
-3.2-
3
G can also be thought of as the "goal" set.
B does not.
The following result is proved in
Lemma 3.1
[3],
(
Lemma 11).
[
Note that
B
n
G =
ij>
3]:
If x
K, then x is adjacent to at most two
e
other simplices in K.
With the above lemma in mind, we can construct paths of simplices in
Let Cx^>. be a maximal sequence of simplices in K such that x. ~ x
K.
and x.
,
±
x +i
-
If X 4
•
i-
s
left endpoint of this sequence, and x
there exists a unique simplex x._ 1
x.
,
to the sequence.
and x.
<=
x
.
,
such that x._ 1
e
i
G,
.
and we append x.
.
,
then
B, and we append
Likewise, if x. is a right endpoint of the sequence,
then there exists a unique simplex x..- c x., such that x.
i G,
..
_
e B,
The new sequence, with possible
to the sequence.
endpoints added, is a path on K.
We have the following results, which are central to the proofs the
combinatorial theorems in the next section:
Lemma 3.2
(
[3],
on K, then x
e
Lemma
[3],
3
.
(
Lemma 12).
Let x
e
K.
If x is an endpoint of a path
B u G.
Lemma 13).
If K is finite, then B and G have the same
parity.
-3.3-
s
4.
.
A Strong Simplotope Theorem, and Extensions
Let C
n
=(xeIRn
< x < e},
set of extreme points of C
,
Let C be a triangulation of C
i.e. Y
n
manifold corresponding to C.
the n-dimensional cube.
Let Y
= {y e IR
i
y. =
with vertex set K
,
n
an element y = £(v) of Y to each vertex of v in K
define RL(x) =
u
RL(v)
K
e
be the
= 1,..., n}.
and let K be the pseudo-
+ Y n be a function that assigns
Let £(•) = K
number of initial zeroes in £(v), for v
or 1,
n
.
,
and let RL(v) be the
Thus
< RL(v)
< n.
For x
e
K,
Kuhn's Strong Cubical Sperner lemma can be stated
.
vex
as follows:
Strong Cubical Sperner lemma (Kuhn [7]).
vertex set K
K
* Y
v.
=
,
Let C be a triangulation of C
and let K be the pseudomanifold corresponding to C.
and v. =
1
implies £.(v) = 1,
there exists an odd number of simplices x
In
[
7 ],
j
with
Let £(•)
be a given vector labelling with the property that for each v
implies £.(v) =
n
= 1,..., n.
e
K
:
,
Then
K such that RL(x) = {0,..., n}
e
Kuhn shows that the Strong Cubical Sperner lemma implies the Brouwer
fixed-point theorem, and that Brouwer'
s
theorem in turn implies a weaker
version of the above lemma.
In this section, we present a combinatorial theorem on the simplotope
that generalizes Kuhn's Strong Cubical lemma, and that implies Sperner'
lemma on the simplex [11].
Define the standard (m S
={xelR
|
e
•
1)
-simplex in
x = 1, x S 0}.
IR
to be the set
Our concern centers on the simplotope
m-.-l
formed by taking the product of n standard simplices, namely S = S
x ...
an"*
where we presume t.. > 1, j = 1,
x S
n, to avoid trivialities.
If v
,
is an element of S,
.
let
v-
1
denote the
j
.
.
,
— concatenated vector of
-4.0-
v,
j
= 1,...,
n,
.
— component of
k— unit vector in
and let v~ denote the k
Define e J
to be the
n
and define M = E (m-
pseudomanifold corresponding to
y
J
Ik.
= eJ
j
= 1,...,
n.
Let £(•)
i
C.
of y
K
:
S
,
JR.
^
,
j
= 1,...
and define E = (e 11
with vertex set K
;
n.
e 21 ;...; e 111 )
£
and let K be the
,
Let Y be the set of extreme points of S,
for some k
f
each concatenated vector
k = 1,..., m.,
1
i.e. M is the dimension of S.
- 1),
j=l
Let C be a triangulation of
1i
v-
{1,..., m.},
e
j
= 1,..., n}.
Y is itself an extreme point of
Note that
m i _1
for
S
Y be a vector-labelling function defined on K
•*
As in the Strong Cubical Sperner lemma, for each v
K
e
,
we reduce the vector
label y = £(v) to an integer label L(v) by defining the reduced label
A
L(v) =
£
£ J (v)),
leading zeroes in
(//
"*
i
set {1,..., n} such that % (v) = e
n=3,
m,
= 4, m„ = 5,
and m^
4.
i» m i
where
j
is the largest index
for all i
< j.
in the
j
For example, suppose
Then the reduced label construction is
shown below for several vector labels.
A
Uv)
(0,
0,
0,
1;
0,
1,
0,
0,
0;
0,
1,
0,
0)
(1,
0,
0,
0;
0,
0,
0,
1,
0;
0,
1,
0,
0)
(0,
0,
0,
1;
0,
0,
0,
0,
1;
0,
1,
0,
0)
(0,
0,
0,
1;
0,
0,
0,
0,
1;
0,
0,
0,
1)
Note that for any £(v)
€
Y,
Strong Simplotope Theorem
with vertex set K
£.(•)
£^(v)
K
:
>
->
0,
Our main result is the following:
i^-l
m -1
x...x S
Let C be a triangulation of S = S
<,
L(v) < M.
and let K be the pseudomanifold corresponding to
,
Let
C.
Y be a vector labelling function with the property that if
then v^
>
0,
reduced labelling of v.
k = 1,..., m.
,
j
1,..., n, and let L(*) be the
Then there exists an odd number of simplices x
such that L(x) = {0,..., M}
-4.1-
e
K
.
,
s
Note that the condition that ^(v)
implies v?
>
,
ensures that £(•)
>
is in some sense "proper" in a manner similar to Sperner's lemma or Rutin'
Strong Cubical Sperner lemma.
Also note that if x
e
K and L(x) = {0,..., M}
x must be an M-simplex, and so all x for which L(x) = {0,..., M) have the
same dimension as the simplotope S.
3
of
which assest the existence of a suitably labelled simplex x
5 ],
[
This is in contrast to theorems 1, 2, and
K
e
whose dimension can be much less than M.
Proof of theorem
The proof of this theorem proceeds by first defining a
:
V-complex associated with K.
is shown that B =
it
Next, the special sets B and G are examined, and
By lemma 3.3, G must have an odd number of elements.
{<}>}.
Finally, it is shown that G = {x
Let N = {0,
i
>
7
=
.
implies (i -
U,
(0},
.
.
M}
,
1)
and let
,
T}
e
= {T c N
J
is the collection:
(M - 1)
|
and
k T,
2},..., {0, 1, 2,..., M - 1}}.
1},
{0,
1,
For T ejf and T^<J>,
|t|
= t >
{0,
L(x) = {0,..., M}}, completing the proof.
|
f
Thus
.
K
e
and T = {0,
.
.
J-l
.
,
t - 1}
,
i
Define
e
T and
A(4>)
= {{E},
whereby there exists
J
a unique index j such that
E (m^ - 1) < t <
Z (m. - 1); upon setting
j-l
i-1
i=l
- 1)
we have
< s < m. - 1.
s = t E (m
Define A(T) to be the pseudoi
J
i=l
manifold corresponding to the restriction of C to the set
,
{v
e
S
i
|
£
v
= e
1
£
'
for £
i
>
j
j-l
dimension
E
,
vr
=
for k
k
>
-l)+s+l-l=t=|T|.
(m ±
s
+
1}
Then this region has
.
It then follows that K, N,
,
i=l
A(*) constitute a V-complex.
Suppose v
If T 4
<)>,
then
e
|t|
for some T
A(T)
e
J
If T =
.
= t > 0, and T = {0,
.
.
j-l
those indices such that
<
t
i=l
Then one of the first
\r
= e
'
,
s
+
1
t
,
- 1}
.
Define
<
E
(m
i
- 1)
s =
,
t
-
E
(m i - 1).
i=l
i=l
components of v^ must be positive, and if
and hence L(v) <
E
(m i - 1) + s =
i=l
-4.2-
0.
and s to be
j-l
j
j
(m i - 1)
E
.
then v = E, and L(v) =
4>,
t
.
j
<
n,
Therefore, for any
and
fj>}
.
T
el,
if t =
|t|,v
implies L(v) S
A(T)
e
t.
Upon examining the set B, note that
Suppose
0-simplex.
and
j
and
s
$
x
5*
Since x
as before.
indices (£, k)
with (£,
,
B
since
,
Then L(x) = T x and x
B.
e
e
<|>
A(<j>)
8*
e
A(T X )
.
(j
,
s)
,
and v
£-1
for all
vex.
If £
j-1
T x = {0,...,
E
<
j
,
then
t =
Define
(m ± - 1)
E
+ k -
1
I
i-1
(m^ - 1) + s - 1}, a contradiction.
I
Since
Suppose x
Tx =
£
I
B|
G,
then
If £ = j,
j-1
and define
=
t
I
Tx
then T x c L(x) c T x
;
l
.
- 1),
whereby x
G.
e
Therefore, G = {x
u
{t}
e
J, whereby
Conversely, suppose L(x) = {0,..., M}
Then x is an M-simplex, and hence T x = {0,..., M-l}
,
s
= 1, G must have an odd number of elements, by lemma 3.3.
{0,..., M -1}, and L(x) = {0,..., M}
L(x) l°J
e
K
|
.
Thus L(x) a T x and
L(x) = {0,..., M}}, and since
G has an odd number of elements, the theorem is proved. B
Note that an algorithm for finding an element x
e
K for which
L(x) = {0,..., M} consists of following the path of adjacent simplices whose
left endpoint is x =
4>
e
The path's right endpoint must be an element of G,
B.
i.e. a simplex with the labels {0,
.
.
.
,
M}
.
The Strong Simplotope theorem is equivalent to Brouwer's fixed-point
theorem,
stated below for the simplotope:
S -> S be a continuous function
Let f(-)
Brouwer's theorem on the simplotope
mn -1
m l -1
Then there exists v e S such that f(v) = v.
x...x S
where S = S
:
:
.
Proof of Brouwer'
s
theorem from the Strong Simplotope theorem: Let f(*)
be a given continuous mapping.
K
,
,
=
and
1,
|
S
= {0,...,
Z (m ± - 1) +
Z (m ± - 1) + k - 1 k L(x) = T
x
i=l
i=l
Thus B = {<(>}.
a contradiction as well.
s -
Tx
L(x) =
i-1 j-1
k <
|
8'A(T X ), there exists an ordered pair of
e
lexicographically less than
k)
contains only one
Let C be a triangulation of
and let K be the pseudomanifold corresponding to C.
-4.3-
S
->
S
:
S
with vertex set
For each v
e
K
,
,.
define £(•)
+ Y by the rule that
K
:
and frXv) - v n» f° r
constructed.
= !>•••» n
1
Z^ (v)
for some
'
where v?
k,
k
>
Such a vector labeling can always be
-
Because the labeling satisfies the hypothesis of the Strong
Simplotope theorem, there exist a simplex x
Let v
= e^
k for which L(x) =
e
{0,..., M}
be any limit point of a subsequence of simplices x for a sequence of
triangulations whose mesh goes to zero.
A continuity argument then implies
that f(v*) = v*. H
Proof of the Strong Simplotope theorem from Brouwer's theorem
given triangulation of
corresponding to
v^
>
Let
0.
Let £(•)
C.
=
£
with vertex set K
S
min
{vjJ.
VjJ.
reduced label of v under £(•)
K
:
-*
If
•
t
and let K be the pseudomanifold
N be given, such that SL^(v)
For each v
0}.
>
Let C be a
:
= 0,
K
e
define f(v)
j-l
t =
let
,
implies
>
L(v)
=v-ee
1
'
1
the
,
+
1
£ e
111
'
!.
J
there is a unique index j such that Z (m^ - 1) < t < Z (m^ - 1)
j-l
i=l
i=l
<
<
and set k = t - Z (n^ - 1)
Then 1
k
m
1
Futhermore tf (v) =
and
If
t
>
0,
.
^k+l
j
<
= X if k <
(v)
.
.
i=l
.
.
m
j
"
ly
and
*k
(v ^
=
and
°
l
Cv +
e
eJ
J
'
v +
£
eJ
'
v +
e
e Jj
>
'
L
I
f(v) =
<
1.
k
-
e
e
J
>
k+l
.
- £ e-
1
k
-£e
Ji
k
k+l
.
<
k =
if
'
,
'
,
r-
if
'
extending
,
containing v
t
e
m-
-
- 1,
and
-,
1
m^-l
j
,
k = m. - 1 and
j
,
,
if
e
S
L(x)
we must have
1) £ L(x)
(t
,
.
If t = 0,
- 1) e L(x)
then we must have
If
.
whereby L(x)
=
f
:
n.
= n.
S
{0,
<
.
.
t
.
,
<
M}
(t
+
1) €
K
-*
S is
.
M, we must have
.
-4.4-
B
Since L(x) ^
L(x)
(t
.
Upon
e
Now let a be the smallest simplex of
.
and let x be the set of vertices of a.
,
.
<
for all v
in a PL fashion over all of S, we have that
f
and so has a fixed point v
+
v ) = ! if k = m.
and
A case by case analysis reveals that f(v)
(t
(
Thus define
n.
exists
+
l
continuous,
C
<j>,
If t = M,
- 1) e L(x)
and
there
then
.
In much the same way that Sperner's lemma implies the Knaster-Kuratowski-
Mazurkiewicz covering lemma on the simplex, the Strong Simplotope theorem
implies a related result:
Simplotope Covering Theorem
n+1
C,
k=
Let C?,
1,..., m. -
i
a
=
C
u
i
,
m'
= 1,..., n,
and C
o
m
1,..., n, and
= 1}.
.vj.
^n
x...x
S
= S, where we define m
S
,
= 2.
...
n+1
m<m £
Suppose that for all
j-1
j
= 1,...,
Then
n
TJ
n,
n C
cJ.
n^},
c {1
n+
i
-C^^
ker
{v
U
e
Cm
.
J
|
L
<{>
j=l,...,n
keTJ
k=l,
.
.
.
,mj-l
This theorem asserts that the intersection of M +
closed sets that cover
2,
=
mi _ l
mj
1,
j
be M + 1 closed sets whose union is the simplotope S =
and define CJ
E
1,
and
3
of
[5],
S
specially configured
1
This contrasts the covering lemmas
must be nonempty.
which assert that one of n intersections of certain
subsets of (M + n) specially configured closed sets covering
nonempty.
S
must be
Just as the Strong Simplotope theorem has a "full-dimensionality"
property to it, so does the Simplotope Covering theorem in that only M +
1
sets are utilized, and the intersection of all of these sets is nonempty.
The major configuration requirement of the Simplotope Covering theorem
is the requirement that
C? a
u
k
k £ T3
{v
e
C
J
~
mj-l
E
.
keT^
v\
k
= l}.
This requirement
is analogous to the requirement of the covering lemma of Knaster et
says that
u
C
k
3
{v
e
S
|
E
v
k
= 1}
for each T c {1,..., m}
keT
keT
however, the condition pertains not to all of
sets C
ml
.
S,
.
.
al., which
Here,
but to the specially defined
These sets are telescoping unions, as illustrated in the two
examples of Figures 1 and
2,
-4.5-
(0,0,1;0,1)
(1,0,0;0,1)
(0,1,0;0,1)
j=2
LC7j-j
(1,0,0;1,0)
C
3
2
(0,1,0;1,0)
= C2
ct =
l
UC 13
'
c:
S
= S
2
x S
1
2,
m.
=
Figure 1
3,
iru
(1,0;0,1;0,1)
(o,i;0,i;0,i)
J-3
(1,0;1,0;0,1)
*
J-2
j-1
y
ci.oa.oa.o)
1
^
(0,1;0,1;1,0)
(0,1;1,0;1,0)
C
= C2
2
^2
l
U C
3
l
U C
4
l
= C 3 u C\
c3-c
S
= S
1
x S
1
x S
1
;
n = 3,
m;L
Figure
2
= 2, m
2
= 2,
m
3
= 2
The proof follows from an application of the Strong
Proof of theorem:
Simplotope theorem.
Let C be a triangulation of
with vertex set K
S
let K be the pseudomanifold corresponding to C.
mn m-i-1
x...xIR
by the rule:
£(v) e TR
For each v
and
,
define
e
K
,
m
,
v
-'-
-i
(v)
£,
J
= <
1
if vj >
1
if v~
and v
i
ri
e
and for some
>
< j
£
m
,
<
a
>0
and v
e C
else,
k = 1,..., m
k = 1,
.
.
.
•
m.
,
,
j
= 1,..., n.
Then note that &^(v)
,
j
= 1,
Furthermore, note that 1^ (v) ^
.
.
.
This is clearly true for
ft
(v)
X,
<
j
£
m
m^p
,
<
and hence vm
v
e
2
± 0,
C^~
.
v
>0
m
But C^ T_
.
1
3
Thus I (v) ± 0,
1? (v)
= eJ
implies
vj*
j
j_1
£
(v)
e
C
m
=
1
.
,
£
>
0,
but that
C
Thus
i =
u
m
.
1,...,
SL
j
and v
C^.,
k=l,...,mj
= 1,...,
n.
k = 1,..., m
.
,
j
iHv)
= e
(v)
= 0.
'
- 1,
x
i =
,
1,..., i-1,
In particular, v^._
e
C^.,
U
i.e.
Then for all
- 1.
whereby
P (v)
and
>
t 0.
V >Q
^
Define £(v) = K°
V (v)
for some k for which
'
4 0,
= 1,..., n.
j
j
for all
>
v^.
0,
Suppose it is true for 1,...,
implies v
and v
>
= 1.
j
± 0,...,
(v)
,
n.
,
implies
>
>
0.
= 1,..., n.
Y by the rule that
-*
Thus
I
3
(v)
Y,
e
and ^(v)
>
Therefore £(•) satisfies the
hypothesis of the Strong Simplotope theorem, and thus there exists a simplex
x
e
K such that L(x) = {0,..., M}
.
be a limit point of a subsequence
Let v
of an infinite sequence of such x for asequence of triangulations C of
mesh goes to zero.
j
= 1,..., n, and C,
Then since each set
"
is closed, v
is closed k = 1,..., m.
Cr.
k
whose
- 1,
j
C~
n
€
j=l,
k=l,
S
.
.
.,n
.
.
.m.j-1
n
C.
,
proving the theorem. H
Paralleling the covering theorem of Knaster et. al., we have:
The Simplotope Covering Theorem implies Brouwer's theorem
f
:
S
+
S
be a given continuous function.
-4.6-
:
To see this, let
Define the following sets:
,
:(veS|
cl
k
c£
{v
S
e
Svjj'.k-l
f*(v)
(v)
f
I
vj.
;
fj (v)
,
:!
±
fc
k=l,...,
n.,-1,
vm± ,
1
1
!}, j-2
]
n,
— 1,
m.
3
,n+l
C^
= {v
S
e
and define C~.
m =
|
u
J
f^(v)
< v^.
C™
= 1,
i
,
£
£>j
i
,
.
.
.
= 1,...,
where m
n,
,
n},
= 2.
n+1n
,
m<m£
First observe that cj. = {v £ S
fj (v)
|
To see this, observe that if v
v^
=s
±
±
i
Qi.,
m ' then v
e
i-1
,
whereby
i =
1,
.
.
fm^Cv)
>
.
j
,
vm
If
.
i = 1,
,
for some i
.
>
we must have ff (v) < vf
CJL = {v e S
j
S =
.
.
.
,
j
,
i=
= 1
j
.
.
.
1,...,
- 1,
£
C^ c
e
1,..., j},
k=l,...,m-j
k=l,
Secondly, observe that if T J c {1
.
m. - 1},
.
,m-j
w
= 1},
J
.C k =
u
{v e S
u
=
{v
u
keT
TJ
,
e
S
Then
e
£
>
j
and
{1,..., m- - 1},
u
C,
then
u
Ck
=>
{v
e
Cj},~
j_1
keTJ
To see this, note that
<
f,(v)
|
vj,
f m± (v)
e
C~j.
j
whereby (v
£
<
j-U
1-1
vm .,
sets,
Z
e
4j-1
=
}
u
keTJ
1
{v e cj?
J" 1
= 1> we must have f^( v )
.^
|
f,(v)
k
<
v,}.
k
for one of the
^ vr
keTJ
cj
J
Thus the sets
1
k
with
-1
v
< vj
*
J
But for any v
e
fj.(v)
|
So suppose that
.
i.
vmi ,
keT J
keTJ
k
....
= 1,..., n.
j
.
Cj,.
<
-1
2
Z
m„£'
Therefore,
C?
u
j=l,...,n
.
<
Thus
.
j=l
keTJ
c
'
= 1,..., n.
j
and m
> Ji
£
and for some k
Cj^.
1,..., n.
Now suppose f^Cv)
n.
,
be the smallest such
C,, and so S is covered by
U
,
for some
m
n, v £ C
1
£
whereby v
i =
for
,
Let
.
i =
,
< vj^,
f mi
|
,
.
vm
<
f mi (v)
Thus fm (v) < v mi
i
< n.
£
< vm±
f mj[ (v)
£
C
e
j
j},j
d,
j
|
1
-
Z
=
vj|.
1} c
u
.c£.
k € TJ
keTJ
= 1,..., n,
k = 1,..., m.
-,,
and
C,
,
being closed
satisfy the hypothesis of the Simplotope Covering theorem, and so there
exists v
e
S
for which v
C
n
e
= l,
k=l,
j
.
.
.
,n
.
.
.
,
n C,
.
Therefore, f(v)
< v,
m^— 1
since e*f(v) = e«v, f(v) = v, proving Brouwer's fixed-point theorem. H
-4.7-
but
.
.
5.
.
Applications to the Simplex and Cube
the Strong Simplotope theorem and the Simplotope Covering
In this section,
theorem are applied to two special simplotopes - the simplex and the cube.
On the
simplex these two theorems specialize to Sperner's lemma [11] and the
Knaster-Kuratowski-Mazurkiewicz Covering lemma [6
On the cube, the Strong
].
Simplotope theorem specializes to Kuhn's Strong Cubical Sperner lemma, and
the Simplotope Covering theorem specializes to a new covering lemma on the
cube.
We first treat the application to the simplex.
(m - 1) -simplex in 1R
and M = m - 1.
Then S
.
Let
S
m~
be the standard
is a simplotope S where n = 1 and m,
= m,
Sperner's combinatorial lemma on the simplex can be stated as
follows:
Sperner's lemma [11].
Let C be a triangulation of
S
and let K be the pseudomanifold corresponding to C.
Let L(»)
be a labelling function with the property that if v
v,
{1,
>
.
0.
.
.
,
with vertex set K
,
e
•*
and L(v) =
K
Then there exists an odd number of simplices x
K
:
e
{1,..., m}
k,
then
K such that L(x) =
m}
This lemma is an instance of the Strong Simplotope theorem.
define
,
«,(•)
:
K°
->
property that V(v)
1
Y by the rule that
£
(v)
= e
1 '1
implies L(v) = k implies
>
^.
>
v,
the hypothesis of the Strong Simplotope theorem.
To see this,
Then £(•) has the
0,
and so £(•) satisfies
Thus there exists an odd
number of simplices whose reduced labels L(x) equal {0,..., M}
{0,..., m - 1}
=
But these simplices are precisely those simplices for which L(x) = {1,..., m}
The covering lemma of Knaster et
.
al. on the simplex can be stated as:
Knaster-Kuratowski-Mazurkiewicz Covering lemma
[
6
]
:
Let C
,
.
.
.
,
C™ be m
m
closed sets such that
_.
_.
1
= S
u C
,
and such that for any T c {1,..., m}, T ?
k=l
-5.0-
<{>,
.
.
^
C
u
{v
S
£
m
- 1}.
E v,
I
keT
,
.
Then
C
n
keT
±
<j>
k=l
This lemma is an instance of the Simplotope Covering theorem, where
C
k
= C
C^,
k = 1,
,
.
.
,
.
.
and C^
= C
= C
mi
m
then follows that the sets
It
.
,
nC m
nC,
and thus
- 1,
satisfy the hypothesis of the Simplotope Covering theorem,
m
k = 1,..., m,
m-1
m
k
=
C
n
^
d>
k=l
k=l
In applying the theorems of section 4 to the cube, note that the n-cube,
= {v
defined as C
m^-1
S
el
*%"!
= S
x.
.
.x S
where m. =
,
2,
define its corresponding v = h(w)
each v
j
e
C
= 1,...,
,
:
K
->
n, and
by v. = wj,
C
its corresponding w = h
(v)
vector
x
e
e
ii(v),
= 1,..., n.
e
S,
Then for
w^ = 1 - v.
,
n
Let C be a triangulation of C
:
and K be the pseudomanifold corresponding to C.
,
{y
7 ])
[
C
e
=
y.
or
1,
= 1,
j
.
.
.
,
K
,
with
Let
n} be a vector-labeling function
for which v. = 1 implies £.(v) = 1 and v. =
for each v
For w
is given by \A = v.
S
e
j
M = n.
Kuhn's strong cubical lemma is stated as:
n.
vertex set K
= 1
j
e
Strong Cubical Sperner Lemma (Kuhn
£(•)
is isomorphic to the simplotope
< v < e}
implies £.(v) =
0,
j
= 1,..., n,
and define L(v) to be the number of initial zeroes of the
for each v
K
e
Then there exists an odd number of simplices
.
K for which L(x) = {0,..., n}
To see how the strong Cubical Sperner lemma follows from the Strong
m^-1
Simplotope theorem, define
S =
x.
S
and the linear functions h(*) and h
induces the triangulation C of
S
.
(•)
.x S
mn -
-'-
where
,
as above.
m.
(v)
for a
unique v
K
e
C e^' 1
,
whose vertex set is K
if
and define £(w)
J>
.(v)
= 1
3
IjCw) =
<
e
j
'
2
if
Ji.(v)
J
-5.1-
=
.
e
= 1,..., n,
j
The triangulation C of C
= h~
pseudomanifold K induces a pseudomanifold K in like manner.
w = h
= 2,
(K
)
,
n
and the
For each w
Y by the rule that
e
K
,
* Y has the property that
Then £(•) = K
£r.(w)
implies w?
>
satisfies the hypothesis of the Strong Simplotope theorem.
exists an odd number of simplices x
is the reduced labeling of 2,(0
L(«)
L(v) =
x
£
j
where v = h
(w)
K
e
K for which L(x)
e
But for w
.
e
K
,
=
>
0,
and so
Thus there
{0,..., n]
L(w) =
j
,
where
if and only if
Thus there exists an odd number of simplices
.
K for which L(x) = {0,..., n}
,
proving the lemma.
We also have the following:
be (n + 1) closed sets such
C
Let C ,..., C
Cubical Covering lemma
n+l_.
J =
and such that for each j = 1,..., n,
C
that
u C
n+1 _n+1 _.
j=l n+l_.
J
J
v. = 0}.
u C
u
C J => {v e
v. = 1} and
u C
C J => {v e
J
2
i=j
i=j+l
i=j
n+l_.
Then n C1 i
j=l
,
:
,
4> .
This lemma follows from the Simplotope Covering theorem by letting
= S
S
C
J
'
mi-1
=
{h
""n
x...x
S
(v)
e S
-1
where
,
J
v
e
n
The sets C ',..., C
n+1
theorem, and thus n C"
j-1
is illustrated in Figure
'
.
1
'
CJ
}
m.
j
,
= 2,
j
= 1,..., n,
= 1,...,
n+1,
and by defining
where h~
(•)
is defined above.
satisfy the hypotheses of the Simplotope Covering
n+1
This cubical covering lemma
whereby n C" ^
f
j-l
.
1
<b
4> .
,
3.
-5.2-
(1,1,1)
1
c
(1,1,0)
1,0,0)1^
(1,0,0)
Cubical Covering Theorem, n =
Figure
3
3
.
6
.
Concluding Remarks
The Strong Simplotope theorem has been shown in section
5
to imply
Sperner's lemma on the simplex and Kuhn's Strong Cubical Sperner lemma.
In
[5], another combinatorial theorem on the simplotope is presented, which also
implies Sperner's lemma on the simplex.
However, this theorem implies a
different result on the cube, which assests the existence of 1-simplex with
certain labels,
as opposed to an n-simplex with labels {0,..., M}
-6.0-
References
[1
]
"Fixed-Point and Related Theorems for Non-Compact Sets,"
Fan, Ky,
Game Theory and Related Topics Eds: 0. Moeschlin and D. Pallaschke,
North-Holland Publishing Company, 1979.
,
[2]
"Variable Dimension Complexes with Applications,"
Freund, R.M.
doctoral dissertation, Department of Operations Research, Stanford
University, Stanford, CA., 1980.
[3]
"Variable Dimension Complexes, Part
Mathematics of Operations Research to appear.
,
I:
,
Basic Theory,"
,
[4]
"Variable Dimension Complexes, Part II: A Unified
Approach to Some Combinatorial Lemmas in Topology," Mathematics of
Operations Research to appear.
,
,
[5]
"Combinatorial Theorems on the Simplotope that
Generalize Results on the Simplex and Cube," Working Paper No. 1556-84,
Sloan School of Management, M.I.T., submitted to Mathematics of
Operations Research
,
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