Compact convex set
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Let $K$ be a convex set in $mathbb{R^n}$
a) For arbitrary $x_1,x_2,...,x_{n+1}in K$ prove that
intersection of all sets $frac{1}{n}x_i+K$ is nonempty for all $i=1,2,...,n+1$.
b) If $K$ is compact set, prove that there exists $xin mathbb{R^n}$ such that $x-frac{1}{n}K subseteq K$
In a) part I tryed to use induction to prove the claim but not sure if it is correct way. Is there any other posibility to prove?
In part b) I am not sure how the assumption that $K$ is compact can be used? Should I conclude something about b) part based on the a) part?
convex-analysis convex-hulls
asked Nov 18 at 12:31
XYZ
978
add a comment |
up vote
1
down vote
favorite
Let $K$ be a convex set in $mathbb{R^n}$
a) For arbitrary $x_1,x_2,...,x_{n+1}in K$ prove that
intersection of all sets $frac{1}{n}x_i+K$ is nonempty for all $i=1,2,...,n+1$.
b) If $K$ is compact set, prove that there exists $xin mathbb{R^n}$ such that $x-frac{1}{n}K subseteq K$
In a) part I tryed to use induction to prove the claim but not sure if it is correct way. Is there any other posibility to prove?
In part b) I am not sure how the assumption that $K$ is compact can be used? Should I conclude something about b) part based on the a) part?
convex-analysis convex-hulls
asked Nov 18 at 12:31
XYZ
978
a) Start studying the sequence of points in $K$, if you look $frac{x_i}{n}in K$, $forall i$.
– james watt
Nov 18 at 12:41
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $K$ be a convex set in $mathbb{R^n}$
a) For arbitrary $x_1,x_2,...,x_{n+1}in K$ prove that
intersection of all sets $frac{1}{n}x_i+K$ is nonempty for all $i=1,2,...,n+1$.
b) If $K$ is compact set, prove that there exists $xin mathbb{R^n}$ such that $x-frac{1}{n}K subseteq K$
In a) part I tryed to use induction to prove the claim but not sure if it is correct way. Is there any other posibility to prove?
In part b) I am not sure how the assumption that $K$ is compact can be used? Should I conclude something about b) part based on the a) part?
convex-analysis convex-hulls
asked Nov 18 at 12:31
XYZ
978
Let $K$ be a convex set in $mathbb{R^n}$
a) For arbitrary $x_1,x_2,...,x_{n+1}in K$ prove that
intersection of all sets $frac{1}{n}x_i+K$ is nonempty for all $i=1,2,...,n+1$.
b) If $K$ is compact set, prove that there exists $xin mathbb{R^n}$ such that $x-frac{1}{n}K subseteq K$
In a) part I tryed to use induction to prove the claim but not sure if it is correct way. Is there any other posibility to prove?
In part b) I am not sure how the assumption that $K$ is compact can be used? Should I conclude something about b) part based on the a) part?
convex-analysis convex-hulls
convex-analysis convex-hulls
asked Nov 18 at 12:31
XYZ
978
asked Nov 18 at 12:31
XYZ
978
edited Nov 20 at 17:24
asked Nov 18 at 12:31
XYZ
978
asked Nov 18 at 12:31
XYZ
978
asked Nov 18 at 12:31
XYZ
978
978
a) Start studying the sequence of points in $K$, if you look $frac{x_i}{n}in K$, $forall i$.
– james watt
Nov 18 at 12:41
add a comment |
a) Start studying the sequence of points in $K$, if you look $frac{x_i}{n}in K$, $forall i$.
– james watt
Nov 18 at 12:41
a) Start studying the sequence of points in $K$, if you look $frac{x_i}{n}in K$, $forall i$.
– james watt
Nov 18 at 12:41
a) Start studying the sequence of points in $K$, if you look $frac{x_i}{n}in K$, $forall i$.
– james watt
Nov 18 at 12:41
add a comment |
1 Answer
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(a): The intersection contains $frac1n(x_1+dots+x_{n+1}).$
(b): Use (a) and Helly's theorem to deduce that there exists some $x$ in $bigcap_{kin K}(frac1nk+K).$ Unwrapping notation shows that $x-frac1n kin K$ for each $kin K,$ which is what you want.
answered Nov 22 at 13:44
Dap
14.1k533
Thank you so much for help.
– XYZ
Nov 25 at 11:54
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
(a): The intersection contains $frac1n(x_1+dots+x_{n+1}).$
(b): Use (a) and Helly's theorem to deduce that there exists some $x$ in $bigcap_{kin K}(frac1nk+K).$ Unwrapping notation shows that $x-frac1n kin K$ for each $kin K,$ which is what you want.
answered Nov 22 at 13:44
Dap
14.1k533
Thank you so much for help.
– XYZ
Nov 25 at 11:54
add a comment |
up vote
0
down vote
(a): The intersection contains $frac1n(x_1+dots+x_{n+1}).$
(b): Use (a) and Helly's theorem to deduce that there exists some $x$ in $bigcap_{kin K}(frac1nk+K).$ Unwrapping notation shows that $x-frac1n kin K$ for each $kin K,$ which is what you want.
answered Nov 22 at 13:44
Dap
14.1k533
Thank you so much for help.
– XYZ
Nov 25 at 11:54
add a comment |
up vote
0
down vote
up vote
0
down vote
(a): The intersection contains $frac1n(x_1+dots+x_{n+1}).$
(b): Use (a) and Helly's theorem to deduce that there exists some $x$ in $bigcap_{kin K}(frac1nk+K).$ Unwrapping notation shows that $x-frac1n kin K$ for each $kin K,$ which is what you want.
answered Nov 22 at 13:44
Dap
14.1k533
(a): The intersection contains $frac1n(x_1+dots+x_{n+1}).$
(b): Use (a) and Helly's theorem to deduce that there exists some $x$ in $bigcap_{kin K}(frac1nk+K).$ Unwrapping notation shows that $x-frac1n kin K$ for each $kin K,$ which is what you want.
answered Nov 22 at 13:44
Dap
14.1k533
answered Nov 22 at 13:44
Dap
14.1k533
answered Nov 22 at 13:44
Dap
14.1k533
answered Nov 22 at 13:44
Dap
14.1k533
14.1k533
Thank you so much for help.
– XYZ
Nov 25 at 11:54
add a comment |
Thank you so much for help.
– XYZ
Nov 25 at 11:54
Thank you so much for help.
– XYZ
Nov 25 at 11:54
Thank you so much for help.
– XYZ
Nov 25 at 11:54
add a comment |
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a) Start studying the sequence of points in $K$, if you look $frac{x_i}{n}in K$, $forall i$.
– james watt
Nov 18 at 12:41