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Vojtěch Jarník IMC
2001 VJIMC
Problem 4
A+B subset A+C, A bounded, C closed, convex
A+B subset A+C, A bounded, C closed, convex
Source: VJIMC 2001 1.4
July 15, 2021
set theory
Problem Statement
Let
A
,
B
,
C
A,B,C
A
,
B
,
C
be nonempty sets in
R
n
\mathbb R^n
R
n
. Suppose that
A
A
A
is bounded,
C
C
C
is closed and convex, and
A
+
B
⊆
A
+
C
A+B\subseteq A+C
A
+
B
⊆
A
+
C
. Prove that
B
⊆
C
B\subseteq C
B
⊆
C
. Recall that
E
+
F
=
{
e
+
f
:
e
∈
E
,
f
∈
F
}
E+F=\{e+f:e\in E,f\in F\}
E
+
F
=
{
e
+
f
:
e
∈
E
,
f
∈
F
}
and
D
⊆
R
n
D\subseteq\mathbb R^n
D
⊆
R
n
is convex iff
t
x
+
(
1
−
t
)
y
∈
D
tx+(1-t)y\in D
t
x
+
(
1
−
t
)
y
∈
D
for all
x
,
y
∈
D
x,y\in D
x
,
y
∈
D
and any
t
∈
[
0
,
1
]
t\in[0,1]
t
∈
[
0
,
1
]
.
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