Relation between a convex set and convex function?
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I was reading about convex sets and convex functions. I would like to know if there is a relation between convex functions and convex sets. Like, a function whose domain and range is a convex set is called a convex function or something on those lines. (Note: I don't know whether the above line is true or if it makes any sense. I wrote it to give an idea of what I'm expecting.) I also found a few questions and answers in this site, but they are for some specific examples. I am looking for a general relation.
Any explanation is highly appreciated. Thanks!
linear-algebra convex-optimization
asked Sep 4 at 5:49
Nagabhushan S N
489
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up vote
1
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I was reading about convex sets and convex functions. I would like to know if there is a relation between convex functions and convex sets. Like, a function whose domain and range is a convex set is called a convex function or something on those lines. (Note: I don't know whether the above line is true or if it makes any sense. I wrote it to give an idea of what I'm expecting.) I also found a few questions and answers in this site, but they are for some specific examples. I am looking for a general relation.
Any explanation is highly appreciated. Thanks!
linear-algebra convex-optimization
asked Sep 4 at 5:49
Nagabhushan S N
489
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I was reading about convex sets and convex functions. I would like to know if there is a relation between convex functions and convex sets. Like, a function whose domain and range is a convex set is called a convex function or something on those lines. (Note: I don't know whether the above line is true or if it makes any sense. I wrote it to give an idea of what I'm expecting.) I also found a few questions and answers in this site, but they are for some specific examples. I am looking for a general relation.
Any explanation is highly appreciated. Thanks!
linear-algebra convex-optimization
asked Sep 4 at 5:49
Nagabhushan S N
489
I was reading about convex sets and convex functions. I would like to know if there is a relation between convex functions and convex sets. Like, a function whose domain and range is a convex set is called a convex function or something on those lines. (Note: I don't know whether the above line is true or if it makes any sense. I wrote it to give an idea of what I'm expecting.) I also found a few questions and answers in this site, but they are for some specific examples. I am looking for a general relation.
Any explanation is highly appreciated. Thanks!
linear-algebra convex-optimization
linear-algebra convex-optimization
asked Sep 4 at 5:49
Nagabhushan S N
489
asked Sep 4 at 5:49
Nagabhushan S N
489
asked Sep 4 at 5:49
Nagabhushan S N
489
asked Sep 4 at 5:49
Nagabhushan S N
489
asked Sep 4 at 5:49
Nagabhushan S N
489
489
add a comment |Â
add a comment |Â
1 Answer
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If $f$ is convex, then the set $(x,y):y>f(x), aleq xleq b$ is convex.
answered Sep 4 at 5:52
Przemysław Scherwentke
11.8k52751
So, a convex set is defined from (using) a convex function. Is that right?
– Nagabhushan S N
Sep 4 at 5:58
@NagabhushanSN No, a convex set is much more general. A set $Q$ is convex if for all $x$, $yin Q$ the interval with ends $x$, $y$ lies entirely in $Q$.
– Przemysław Scherwentke
Sep 4 at 6:00
Consider for example the set $(x,y),0le x le1, 0 le y le 1$,which is convex but cannot be defined using a function from $mathbbR$ to itself.
– nicomezi
Sep 4 at 6:03
1
@PrzemysławScherwentke Okay. So convex sets and convex functions are two separate entities (without any relation between them). I mean you don't need one to define the other. But given a convex function, you can construct a convex set. Am I right?
– Nagabhushan S N
Sep 4 at 6:03
That is basically what his answer tells you.
– nicomezi
Sep 4 at 6:03
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
accepted
If $f$ is convex, then the set $(x,y):y>f(x), aleq xleq b$ is convex.
answered Sep 4 at 5:52
Przemysław Scherwentke
11.8k52751
So, a convex set is defined from (using) a convex function. Is that right?
– Nagabhushan S N
Sep 4 at 5:58
@NagabhushanSN No, a convex set is much more general. A set $Q$ is convex if for all $x$, $yin Q$ the interval with ends $x$, $y$ lies entirely in $Q$.
– Przemysław Scherwentke
Sep 4 at 6:00
Consider for example the set $(x,y),0le x le1, 0 le y le 1$,which is convex but cannot be defined using a function from $mathbbR$ to itself.
– nicomezi
Sep 4 at 6:03
1
@PrzemysławScherwentke Okay. So convex sets and convex functions are two separate entities (without any relation between them). I mean you don't need one to define the other. But given a convex function, you can construct a convex set. Am I right?
– Nagabhushan S N
Sep 4 at 6:03
That is basically what his answer tells you.
– nicomezi
Sep 4 at 6:03
add a comment |Â
up vote
3
down vote
accepted
If $f$ is convex, then the set $(x,y):y>f(x), aleq xleq b$ is convex.
answered Sep 4 at 5:52
Przemysław Scherwentke
11.8k52751
So, a convex set is defined from (using) a convex function. Is that right?
– Nagabhushan S N
Sep 4 at 5:58
@NagabhushanSN No, a convex set is much more general. A set $Q$ is convex if for all $x$, $yin Q$ the interval with ends $x$, $y$ lies entirely in $Q$.
– Przemysław Scherwentke
Sep 4 at 6:00
Consider for example the set $(x,y),0le x le1, 0 le y le 1$,which is convex but cannot be defined using a function from $mathbbR$ to itself.
– nicomezi
Sep 4 at 6:03
1
@PrzemysławScherwentke Okay. So convex sets and convex functions are two separate entities (without any relation between them). I mean you don't need one to define the other. But given a convex function, you can construct a convex set. Am I right?
– Nagabhushan S N
Sep 4 at 6:03
That is basically what his answer tells you.
– nicomezi
Sep 4 at 6:03
add a comment |Â
up vote
3
down vote
accepted
up vote
3
down vote
accepted
If $f$ is convex, then the set $(x,y):y>f(x), aleq xleq b$ is convex.
answered Sep 4 at 5:52
Przemysław Scherwentke
11.8k52751
If $f$ is convex, then the set $(x,y):y>f(x), aleq xleq b$ is convex.
answered Sep 4 at 5:52
Przemysław Scherwentke
11.8k52751
edited Sep 4 at 5:58
answered Sep 4 at 5:52
Przemysław Scherwentke
11.8k52751
answered Sep 4 at 5:52
Przemysław Scherwentke
11.8k52751
answered Sep 4 at 5:52
Przemysław Scherwentke
11.8k52751
11.8k52751
So, a convex set is defined from (using) a convex function. Is that right?
– Nagabhushan S N
Sep 4 at 5:58
@NagabhushanSN No, a convex set is much more general. A set $Q$ is convex if for all $x$, $yin Q$ the interval with ends $x$, $y$ lies entirely in $Q$.
– Przemysław Scherwentke
Sep 4 at 6:00
Consider for example the set $(x,y),0le x le1, 0 le y le 1$,which is convex but cannot be defined using a function from $mathbbR$ to itself.
– nicomezi
Sep 4 at 6:03
1
@PrzemysławScherwentke Okay. So convex sets and convex functions are two separate entities (without any relation between them). I mean you don't need one to define the other. But given a convex function, you can construct a convex set. Am I right?
– Nagabhushan S N
Sep 4 at 6:03
That is basically what his answer tells you.
– nicomezi
Sep 4 at 6:03
add a comment |Â
So, a convex set is defined from (using) a convex function. Is that right?
– Nagabhushan S N
Sep 4 at 5:58
@NagabhushanSN No, a convex set is much more general. A set $Q$ is convex if for all $x$, $yin Q$ the interval with ends $x$, $y$ lies entirely in $Q$.
– Przemysław Scherwentke
Sep 4 at 6:00
Consider for example the set $(x,y),0le x le1, 0 le y le 1$,which is convex but cannot be defined using a function from $mathbbR$ to itself.
– nicomezi
Sep 4 at 6:03
1
@PrzemysławScherwentke Okay. So convex sets and convex functions are two separate entities (without any relation between them). I mean you don't need one to define the other. But given a convex function, you can construct a convex set. Am I right?
– Nagabhushan S N
Sep 4 at 6:03
That is basically what his answer tells you.
– nicomezi
Sep 4 at 6:03
So, a convex set is defined from (using) a convex function. Is that right?
– Nagabhushan S N
Sep 4 at 5:58
So, a convex set is defined from (using) a convex function. Is that right?
– Nagabhushan S N
Sep 4 at 5:58
@NagabhushanSN No, a convex set is much more general. A set $Q$ is convex if for all $x$, $yin Q$ the interval with ends $x$, $y$ lies entirely in $Q$.
– Przemysław Scherwentke
Sep 4 at 6:00
@NagabhushanSN No, a convex set is much more general. A set $Q$ is convex if for all $x$, $yin Q$ the interval with ends $x$, $y$ lies entirely in $Q$.
– Przemysław Scherwentke
Sep 4 at 6:00
Consider for example the set $(x,y),0le x le1, 0 le y le 1$,which is convex but cannot be defined using a function from $mathbbR$ to itself.
– nicomezi
Sep 4 at 6:03
Consider for example the set $(x,y),0le x le1, 0 le y le 1$,which is convex but cannot be defined using a function from $mathbbR$ to itself.
– nicomezi
Sep 4 at 6:03
1
1
@PrzemysławScherwentke Okay. So convex sets and convex functions are two separate entities (without any relation between them). I mean you don't need one to define the other. But given a convex function, you can construct a convex set. Am I right?
– Nagabhushan S N
Sep 4 at 6:03
@PrzemysławScherwentke Okay. So convex sets and convex functions are two separate entities (without any relation between them). I mean you don't need one to define the other. But given a convex function, you can construct a convex set. Am I right?
– Nagabhushan S N
Sep 4 at 6:03
That is basically what his answer tells you.
– nicomezi
Sep 4 at 6:03
That is basically what his answer tells you.
– nicomezi
Sep 4 at 6:03
add a comment |Â
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