Theorem 11
(Vertex-isoperimetric inequality in the grid)
.
Assuming that:
A
⊂
[
k
]
n
C
is the initial segment of
simplicial order
on
[
k
]
n
with
|
C
|
=
|
A
|
Then
|
N
(
C
)
|
≤
|
N
(
A
)
|
. In particular, if
|
A
|
≥
|
{
x
:
|
x
|
≤
r
}
|
then
|
N
(
A
)
|
≥
|
{
x
:
|
x
|
≤
r
+
1
}
|
.