The Image of Extreme Subsets under Linear Maps
$begingroup$
Lax provides the following defintion for extreme subsets:
A subset $E$ of a convex set $K$ is an extreme subset when
$E$ is convex and non empty
Whenever $x in E$ is expressed as $frac{y+z}{2}$, $y,z in K$, then $y,z in E$
He provides two exercises:
If $M$ is a linear map from $M:X rightarrow U$ and $E subset K subset U$ where $K$ is convex and $E$ is an extreme subset. Then the inverse image of $E$ is either empty or an extreme subset of the inverse image of $K$.
Give an example of how the image of an extreme subset under a linear map need not be an extreme subset of the image.
For 1, I can show this -- but I'm having trouble with the "empty" part, why can the inverse image of an extreme subset be empty?
And 2. I'm having trouble producing an example.
Thoughts?
linear-algebra functional-analysis convex-analysis
asked Dec 13 '18 at 15:41
yoshiyoshi
1,256917
$endgroup$
add a comment |
$begingroup$
Lax provides the following defintion for extreme subsets:
A subset $E$ of a convex set $K$ is an extreme subset when
$E$ is convex and non empty
Whenever $x in E$ is expressed as $frac{y+z}{2}$, $y,z in K$, then $y,z in E$
He provides two exercises:
If $M$ is a linear map from $M:X rightarrow U$ and $E subset K subset U$ where $K$ is convex and $E$ is an extreme subset. Then the inverse image of $E$ is either empty or an extreme subset of the inverse image of $K$.
Give an example of how the image of an extreme subset under a linear map need not be an extreme subset of the image.
For 1, I can show this -- but I'm having trouble with the "empty" part, why can the inverse image of an extreme subset be empty?
And 2. I'm having trouble producing an example.
Thoughts?
linear-algebra functional-analysis convex-analysis
asked Dec 13 '18 at 15:41
yoshiyoshi
1,256917
$endgroup$
2
$begingroup$
A trivial example for (1) would be to take $M=0$ and $E = K = {x}$ where $x ne 0$.
$endgroup$
– Nate Eldredge
Dec 13 '18 at 16:00
$begingroup$
oh right, $E$ and $K$ have to live in the image of $M$!
$endgroup$
– yoshi
Dec 13 '18 at 16:03
add a comment |
$begingroup$
Lax provides the following defintion for extreme subsets:
A subset $E$ of a convex set $K$ is an extreme subset when
$E$ is convex and non empty
Whenever $x in E$ is expressed as $frac{y+z}{2}$, $y,z in K$, then $y,z in E$
He provides two exercises:
If $M$ is a linear map from $M:X rightarrow U$ and $E subset K subset U$ where $K$ is convex and $E$ is an extreme subset. Then the inverse image of $E$ is either empty or an extreme subset of the inverse image of $K$.
Give an example of how the image of an extreme subset under a linear map need not be an extreme subset of the image.
For 1, I can show this -- but I'm having trouble with the "empty" part, why can the inverse image of an extreme subset be empty?
And 2. I'm having trouble producing an example.
Thoughts?
linear-algebra functional-analysis convex-analysis
asked Dec 13 '18 at 15:41
yoshiyoshi
1,256917
$endgroup$
Lax provides the following defintion for extreme subsets:
A subset $E$ of a convex set $K$ is an extreme subset when
$E$ is convex and non empty
Whenever $x in E$ is expressed as $frac{y+z}{2}$, $y,z in K$, then $y,z in E$
He provides two exercises:
If $M$ is a linear map from $M:X rightarrow U$ and $E subset K subset U$ where $K$ is convex and $E$ is an extreme subset. Then the inverse image of $E$ is either empty or an extreme subset of the inverse image of $K$.
Give an example of how the image of an extreme subset under a linear map need not be an extreme subset of the image.
For 1, I can show this -- but I'm having trouble with the "empty" part, why can the inverse image of an extreme subset be empty?
And 2. I'm having trouble producing an example.
Thoughts?
linear-algebra functional-analysis convex-analysis
linear-algebra functional-analysis convex-analysis
asked Dec 13 '18 at 15:41
yoshiyoshi
1,256917
asked Dec 13 '18 at 15:41
yoshiyoshi
1,256917
asked Dec 13 '18 at 15:41
yoshiyoshi
1,256917
asked Dec 13 '18 at 15:41
yoshiyoshi
1,256917
asked Dec 13 '18 at 15:41
yoshiyoshi
1,256917
1,256917
2
$begingroup$
A trivial example for (1) would be to take $M=0$ and $E = K = {x}$ where $x ne 0$.
$endgroup$
– Nate Eldredge
Dec 13 '18 at 16:00
$begingroup$
oh right, $E$ and $K$ have to live in the image of $M$!
$endgroup$
– yoshi
Dec 13 '18 at 16:03
add a comment |
2
$begingroup$
A trivial example for (1) would be to take $M=0$ and $E = K = {x}$ where $x ne 0$.
$endgroup$
– Nate Eldredge
Dec 13 '18 at 16:00
$begingroup$
oh right, $E$ and $K$ have to live in the image of $M$!
$endgroup$
– yoshi
Dec 13 '18 at 16:03
2
2
$begingroup$
A trivial example for (1) would be to take $M=0$ and $E = K = {x}$ where $x ne 0$.
$endgroup$
– Nate Eldredge
Dec 13 '18 at 16:00
$begingroup$
A trivial example for (1) would be to take $M=0$ and $E = K = {x}$ where $x ne 0$.
$endgroup$
– Nate Eldredge
Dec 13 '18 at 16:00
$begingroup$
oh right, $E$ and $K$ have to live in the image of $M$!
$endgroup$
– yoshi
Dec 13 '18 at 16:03
$begingroup$
oh right, $E$ and $K$ have to live in the image of $M$!
$endgroup$
– yoshi
Dec 13 '18 at 16:03
add a comment |
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$begingroup$
A trivial example for (1) would be to take $M=0$ and $E = K = {x}$ where $x ne 0$.
$endgroup$
– Nate Eldredge
Dec 13 '18 at 16:00
$begingroup$
oh right, $E$ and $K$ have to live in the image of $M$!
$endgroup$
– yoshi
Dec 13 '18 at 16:03