Caratheodory's theorem applied to a disk
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Here's the statement that was given in my class.
Caratheorody's theorem: Let M $subset$ $mathbb R^n$. Then conv$(M)$ is the set of all convex combinations of at most n+1 points from M.
Am I right to assume that the n+1 points mentioned above are not "fixed", and the correct way to interpret the statement is that, given a point x of M, we can find n+1 points of M whose convex combination is x?
I'm asking since my initial interpretation of the statement is that, given the convex hull of a set M, we can find n+1 points of M such that their convex combination is conv$(M)$. But then this statement would not apply to the example of a disk. If M is the unit disk in $mathbb R^2$, I don't see how the convex hull of three points from this disk could be the convex hull of the whole thing.
convex-geometry discrete-geometry
asked Nov 13 at 20:51
Josh Ng
876
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down vote
favorite
Here's the statement that was given in my class.
Caratheorody's theorem: Let M $subset$ $mathbb R^n$. Then conv$(M)$ is the set of all convex combinations of at most n+1 points from M.
Am I right to assume that the n+1 points mentioned above are not "fixed", and the correct way to interpret the statement is that, given a point x of M, we can find n+1 points of M whose convex combination is x?
I'm asking since my initial interpretation of the statement is that, given the convex hull of a set M, we can find n+1 points of M such that their convex combination is conv$(M)$. But then this statement would not apply to the example of a disk. If M is the unit disk in $mathbb R^2$, I don't see how the convex hull of three points from this disk could be the convex hull of the whole thing.
convex-geometry discrete-geometry
asked Nov 13 at 20:51
Josh Ng
876
2
Given any point $x$ of $text{conv}(M)$, we can find $n+1$ points of $M$ such that $x$ is a convex combination of those $n+1$ points. You are right: the choice of $n+1$ points will in general depend on $x$.
– Robert Israel
Nov 13 at 21:04
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Here's the statement that was given in my class.
Caratheorody's theorem: Let M $subset$ $mathbb R^n$. Then conv$(M)$ is the set of all convex combinations of at most n+1 points from M.
Am I right to assume that the n+1 points mentioned above are not "fixed", and the correct way to interpret the statement is that, given a point x of M, we can find n+1 points of M whose convex combination is x?
I'm asking since my initial interpretation of the statement is that, given the convex hull of a set M, we can find n+1 points of M such that their convex combination is conv$(M)$. But then this statement would not apply to the example of a disk. If M is the unit disk in $mathbb R^2$, I don't see how the convex hull of three points from this disk could be the convex hull of the whole thing.
convex-geometry discrete-geometry
asked Nov 13 at 20:51
Josh Ng
876
Here's the statement that was given in my class.
Caratheorody's theorem: Let M $subset$ $mathbb R^n$. Then conv$(M)$ is the set of all convex combinations of at most n+1 points from M.
Am I right to assume that the n+1 points mentioned above are not "fixed", and the correct way to interpret the statement is that, given a point x of M, we can find n+1 points of M whose convex combination is x?
I'm asking since my initial interpretation of the statement is that, given the convex hull of a set M, we can find n+1 points of M such that their convex combination is conv$(M)$. But then this statement would not apply to the example of a disk. If M is the unit disk in $mathbb R^2$, I don't see how the convex hull of three points from this disk could be the convex hull of the whole thing.
convex-geometry discrete-geometry
convex-geometry discrete-geometry
asked Nov 13 at 20:51
Josh Ng
876
asked Nov 13 at 20:51
Josh Ng
876
asked Nov 13 at 20:51
Josh Ng
876
asked Nov 13 at 20:51
Josh Ng
876
asked Nov 13 at 20:51
Josh Ng
876
876
2
Given any point $x$ of $text{conv}(M)$, we can find $n+1$ points of $M$ such that $x$ is a convex combination of those $n+1$ points. You are right: the choice of $n+1$ points will in general depend on $x$.
– Robert Israel
Nov 13 at 21:04
add a comment |
2
Given any point $x$ of $text{conv}(M)$, we can find $n+1$ points of $M$ such that $x$ is a convex combination of those $n+1$ points. You are right: the choice of $n+1$ points will in general depend on $x$.
– Robert Israel
Nov 13 at 21:04
2
2
Given any point $x$ of $text{conv}(M)$, we can find $n+1$ points of $M$ such that $x$ is a convex combination of those $n+1$ points. You are right: the choice of $n+1$ points will in general depend on $x$.
– Robert Israel
Nov 13 at 21:04
Given any point $x$ of $text{conv}(M)$, we can find $n+1$ points of $M$ such that $x$ is a convex combination of those $n+1$ points. You are right: the choice of $n+1$ points will in general depend on $x$.
– Robert Israel
Nov 13 at 21:04
add a comment |
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Given any point $x$ of $text{conv}(M)$, we can find $n+1$ points of $M$ such that $x$ is a convex combination of those $n+1$ points. You are right: the choice of $n+1$ points will in general depend on $x$.
– Robert Israel
Nov 13 at 21:04