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IMC
1996 IMC
6
6
Part of
1996 IMC
Problems
(1)
IMC 1996 Problem 6
Source: IMC 1996
3/5/2021
Upper content of a subset
E
E
E
of the plane
R
2
\mathbb{R}^{2}
R
2
is defined as
C
(
E
)
=
inf
{
∑
i
=
1
n
diam
(
E
i
)
}
\mathcal{C}(E)=\inf\{\sum_{i=1}^{n} \text{diam}(E_{i})\}
C
(
E
)
=
in
f
{
i
=
1
∑
n
diam
(
E
i
)}
where
inf
\inf
in
f
is taken over all finite families of sets
E
1
,
…
,
E
n
E_{1},\dots,E_{n}
E
1
,
…
,
E
n
n
∈
N
n\in \mathbb{N}
n
∈
N
, in
R
2
\mathbb{R}^{2}
R
2
such that
E
⊂
⋃
i
=
1
n
E
i
E\subset \bigcup_{i=1}^{n}E_{i}
E
⊂
⋃
i
=
1
n
E
i
. Lower content of
E
E
E
is defined as
K
(
E
)
=
sup
{
length
(
L
)
∣
L
is a closed line segment onto which
E
can be contracted
}
\mathcal{K}(E)=\sup\{\text{length}(L) |\, L \text{ is a closed line segment onto which $E$ can be contracted}\}
K
(
E
)
=
sup
{
length
(
L
)
∣
L
is a closed line segment onto which
E
can be contracted
}
. Prove that i)
C
(
L
)
=
length
(
L
)
\mathcal{C}(L)=\text{length}(L)
C
(
L
)
=
length
(
L
)
if
L
L
L
is a closed line segment; ii)
C
(
E
)
≥
K
(
E
)
\mathcal{C}(E) \geq \mathcal{K}(E)
C
(
E
)
≥
K
(
E
)
; iii) the equality in ii) is not always true even if
E
E
E
is compact.
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