𝑓
∘
𝑔
(
x
)
∈
𝑆
for every
x
∈
𝐵
, we must have
𝑓
∘
𝑔
(
x
∗
) =
x
∗
∈
𝑆
.
Therefore,
𝑔
(
x
∗
) =
x
∗
which implies that
𝑓
(
x
∗
) =
x
∗
. That is,
x
∗
is a fixed point of
𝑓
.
Brouwer
=
⇒
noretraction
Exercise 2.132.
2.135
Let Λ
𝑘
,
𝑘
= 1
,
2
, . . .
be a sequence of simplicial partitions of
𝑆
in which the
maximum diameter of the subsimplices tend to zero as
𝑘
→ ∞
. By Sperner’s lemma
(Proposition 1.3), every partition Λ
𝑘
has a completely labeled subsimplex with vertices
x
𝑘
0
,
x
𝑘
1
, . . . ,
x
𝑘
𝑛
.
By construction of an admissible labeling, each
x
𝑘
𝑖
belongs to a face
containing
x
𝑖
, that is
x
𝑘
𝑖
∈
conv
{
x
𝑖
, . . .
}
and therefore
x
𝑘
𝑖
∈
𝐴
𝑖
,
𝑖
= 0
,
1
, . . . , 𝑛
Since
𝑆
is compact, each sequence
x
𝑘
𝑖
has a convergent subsequence
x
𝑘
′
𝑖
.
Moreover,
since the diameters of the subsimplices converge to zero, these subsequences must
converge to the same point, say
x
∗
. That is,
lim
𝑘
′
→∞
x
𝑘
′
𝑖
=
x
∗
,
𝑖
= 0
,
1
, . . . , 𝑛
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 Fall '10
 Dr.DuMond
 Macroeconomics, Topology, x∗, Barycentric Coordinates, Foundations of Mathematical Economics, conv { x????

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