Closed bounded convex set in $mathbb{R}^n$
$begingroup$
Let the set $B subset mathbb{R}^n$ be convex, bounded and closed. We want to show that set $B$ is equal to convex hull of its boundary?
convex-analysis
edited Dec 5 '18 at 15:32
Derek Allums
1,70921031
asked Dec 5 '18 at 15:00
B.i.azB.i.az
11
$endgroup$
add a comment |
$begingroup$
Let the set $B subset mathbb{R}^n$ be convex, bounded and closed. We want to show that set $B$ is equal to convex hull of its boundary?
convex-analysis
edited Dec 5 '18 at 15:32
Derek Allums
1,70921031
asked Dec 5 '18 at 15:00
B.i.azB.i.az
11
$endgroup$
add a comment |
$begingroup$
Let the set $B subset mathbb{R}^n$ be convex, bounded and closed. We want to show that set $B$ is equal to convex hull of its boundary?
convex-analysis
edited Dec 5 '18 at 15:32
Derek Allums
1,70921031
asked Dec 5 '18 at 15:00
B.i.azB.i.az
11
$endgroup$
Let the set $B subset mathbb{R}^n$ be convex, bounded and closed. We want to show that set $B$ is equal to convex hull of its boundary?
convex-analysis
convex-analysis
edited Dec 5 '18 at 15:32
Derek Allums
1,70921031
asked Dec 5 '18 at 15:00
B.i.azB.i.az
11
edited Dec 5 '18 at 15:32
Derek Allums
1,70921031
asked Dec 5 '18 at 15:00
B.i.azB.i.az
11
edited Dec 5 '18 at 15:32
Derek Allums
1,70921031
edited Dec 5 '18 at 15:32
Derek Allums
1,70921031
edited Dec 5 '18 at 15:32
Derek Allums
1,70921031
1,70921031
asked Dec 5 '18 at 15:00
B.i.azB.i.az
11
asked Dec 5 '18 at 15:00
B.i.azB.i.az
11
asked Dec 5 '18 at 15:00
B.i.azB.i.az
11
11
add a comment |
add a comment |
1 Answer
1
active
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votes
$begingroup$
By Krein-Milman,
$$
B = overline{mathrm{conv}(mathrm{ext}(B))}
subseteq overline{mathrm{conv}(partial(B))}
subseteq overline{mathrm{conv}(B)}
= overline{B} = B .
$$
answered Dec 5 '18 at 15:38
FedericoFederico
4,934514
$endgroup$
$begingroup$
Yes, I know that the asked question is much simpler, I'm just being lazy
$endgroup$
– Federico
Dec 5 '18 at 15:43
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
By Krein-Milman,
$$
B = overline{mathrm{conv}(mathrm{ext}(B))}
subseteq overline{mathrm{conv}(partial(B))}
subseteq overline{mathrm{conv}(B)}
= overline{B} = B .
$$
answered Dec 5 '18 at 15:38
FedericoFederico
4,934514
$endgroup$
$begingroup$
Yes, I know that the asked question is much simpler, I'm just being lazy
$endgroup$
– Federico
Dec 5 '18 at 15:43
add a comment |
$begingroup$
By Krein-Milman,
$$
B = overline{mathrm{conv}(mathrm{ext}(B))}
subseteq overline{mathrm{conv}(partial(B))}
subseteq overline{mathrm{conv}(B)}
= overline{B} = B .
$$
answered Dec 5 '18 at 15:38
FedericoFederico
4,934514
$endgroup$
$begingroup$
Yes, I know that the asked question is much simpler, I'm just being lazy
$endgroup$
– Federico
Dec 5 '18 at 15:43
add a comment |
$begingroup$
By Krein-Milman,
$$
B = overline{mathrm{conv}(mathrm{ext}(B))}
subseteq overline{mathrm{conv}(partial(B))}
subseteq overline{mathrm{conv}(B)}
= overline{B} = B .
$$
answered Dec 5 '18 at 15:38
FedericoFederico
4,934514
$endgroup$
By Krein-Milman,
$$
B = overline{mathrm{conv}(mathrm{ext}(B))}
subseteq overline{mathrm{conv}(partial(B))}
subseteq overline{mathrm{conv}(B)}
= overline{B} = B .
$$
answered Dec 5 '18 at 15:38
FedericoFederico
4,934514
answered Dec 5 '18 at 15:38
FedericoFederico
4,934514
answered Dec 5 '18 at 15:38
FedericoFederico
4,934514
answered Dec 5 '18 at 15:38
FedericoFederico
4,934514
4,934514
$begingroup$
Yes, I know that the asked question is much simpler, I'm just being lazy
$endgroup$
– Federico
Dec 5 '18 at 15:43
add a comment |
$begingroup$
Yes, I know that the asked question is much simpler, I'm just being lazy
$endgroup$
– Federico
Dec 5 '18 at 15:43
$begingroup$
Yes, I know that the asked question is much simpler, I'm just being lazy
$endgroup$
– Federico
Dec 5 '18 at 15:43
$begingroup$
Yes, I know that the asked question is much simpler, I'm just being lazy
$endgroup$
– Federico
Dec 5 '18 at 15:43
add a comment |
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