ISyE 7682
Fall 2010 – 2nd Homework
Problem 0.1
Let
C
i
⊂ ℜ
n
i
be a convex set for
i
= 1
,
2. Show that
aff
C
1
×
aff
C
2
= aff (
C
1
×
C
2
)
,
ri
C
1
×
ri
C
2
= ri (
C
1
×
C
2
)
.
Problem 0.2
For a nonempty closed convex
C
⊆ ℜ
n
, show that
C
∞
=
{
d
∈ ℜ
n
:
C
+
d
⊆
C
}
.
Problem 0.3
Let
∅ negationslash
=
C
⊂ ℜ
n
be a convex set. Show that:
C
∞
=
∩
ǫ>
0
cl (
∪
0
<t
≤
ǫ
tC
)
.
Problem 0.4
A polyhedron is a set
C
which can be represented as
C
=
{
x
∈ ℜ
n
:
Ax
≤
b
}
for
some
A
∈ ℜ
m
×
n
and
b
∈ ℜ
n
. Assume that the polyhedron
C
=
{
x
∈ ℜ
n
:
Ax
≤
b
}
is nonempty
and bounded.
Show that
C
∞
=
{
d
∈ ℜ
n
:
Ad
≤
0
}
and that the (possibly empty) polyhedron
{
x
∈ ℜ
n
:
Ax
≤
b
′
}
is bounded for any other right hand side
b
′
∈ ℜ
m
.
Problem 0.5
For a nonempty closed convex set
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 Fall '08
 Staff
 Convex set, C∞, nonempty closed convex

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