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and where the
v
i
are
any
objects for which the expression makes sense is called a
convex combination
of the
v
i
.
1.7.4. Theorem.
Let
v
0
,
v
1
,...,
v
k
be k + 1 linearly independent points.
(1) Every point
w
of aff({
v
0
,
v
1
,...,
v
k
}) can be written
uniquely
in the form
k
k
Â
Â
wv
=
a
,
where
a
=
1
.
ii
i
i
=
0
i
=
0
(2) Every point
w
of the simplex s =
v
0
v
1
···
v
k
can be written
uniquely
in the
form
k
k
Â
a
Â
Œ
[]
wv
=
,
where
a
01
,
,
and
a
=
1
.
ii
i
i
i
=
0
i
=
0
Furthermore, the dimension and the vertices of a simplex are uniquely deter-
mined, that is, if
v
0
v
1
···
v
k
=
v
0
¢
v
1
¢
...
v
t
¢, then k = t and
v
i
=
v
i
¢ after a renum-
bering of the
v
i
¢.
Proof.
Lemma 1.7.3 showed that every point
w
has a representation as shown in (1)
and (2). We need to show that it is unique. Suppose that we have two representations
of the form
k
k
Â
a
Â
¢
wv
=
=
a
v
.
ii
ii
i
=
0
i
=
0
Then
k
k
Â
Â
¢
0ww
=-=
a
v
-
a
v
ii
ii
i
=
0
i
=
0
k
(
)
Â
¢
=
aa
-
v
i
i
i
i
=
0
k
k
Ê
Á
ˆ
˜
(
)
(
)
Â
Â
¢
(
)
+
¢
=
aa
-
vv
-
aa
-
v
i
i
i
0
i
i
0
i
=
0
i
=
0
k
(
)
Â
¢
(
)
=
aa
-
vv
-
i
i
i
0
i
=
0
k
(
)
Â
¢
(
)
=
aa
-
v
-
v
0
.
i
i
i
i
=
1
The second to last equality sign follows from the fact that
