$bigcap_{i in I}overline{co}(A_i)=overline{co}left(bigcap_{i in I}overline{A_i}right)$
$begingroup$
Let $(A_i: i in I)$ be a family of sets in a topological vector space such that for all $i,j in I$ there exists $k in I$ for which $A_i cap A_j=A_k$. Denote by $overline{co}(X)$ the closure of the convex hull of $X$. Is it true that
$$
bigcap_{i in I}overline{co}(A_i)=overline{co}left(bigcap_{i in I}overline{A_i}right),,,?
$$
One inclusion: Since $overline{A_i} subseteq overline{co}(A_i)$ for all $i$ then $bigcap_i overline{A_i} subseteq bigcap_i overline{co}(A_i)$. Therefore
$$
overline{co}left(bigcap_i overline{A_i}right) subseteq overline{co}left(bigcap_i overline{co}(A_i)right)=bigcap_i overline{co}(A_i).
$$
What about the other?
Comment 1. The hypothesis on the family ${A_i:i in I}$ is necessary, see Kavi's answer.
Comment 2. The claim is easily seen to be verified if ${A_i: i in I}$ has a minimum, which holds, e.g., if $I$ is finite.
Comment 3. The claim $bigcap_{i}overline{co}(A_i)=overline{co}left(bigcap_{i}A_iright)$ (i.e., replacing $overline{A_i}$ with $A_i$ in the right side) is false. As a counterexample, let $A_i=(0,2^{-i})cup{1}$ for all integers $ige 1$. Then the left side is $[0,1]$ and the right side is ${1}$.
general-topology vector-spaces convex-hulls
asked Dec 21 '18 at 0:23
Paolo LeonettiPaolo Leonetti
11.5k21550
$endgroup$
add a comment |
$begingroup$
Let $(A_i: i in I)$ be a family of sets in a topological vector space such that for all $i,j in I$ there exists $k in I$ for which $A_i cap A_j=A_k$. Denote by $overline{co}(X)$ the closure of the convex hull of $X$. Is it true that
$$
bigcap_{i in I}overline{co}(A_i)=overline{co}left(bigcap_{i in I}overline{A_i}right),,,?
$$
One inclusion: Since $overline{A_i} subseteq overline{co}(A_i)$ for all $i$ then $bigcap_i overline{A_i} subseteq bigcap_i overline{co}(A_i)$. Therefore
$$
overline{co}left(bigcap_i overline{A_i}right) subseteq overline{co}left(bigcap_i overline{co}(A_i)right)=bigcap_i overline{co}(A_i).
$$
What about the other?
Comment 1. The hypothesis on the family ${A_i:i in I}$ is necessary, see Kavi's answer.
Comment 2. The claim is easily seen to be verified if ${A_i: i in I}$ has a minimum, which holds, e.g., if $I$ is finite.
Comment 3. The claim $bigcap_{i}overline{co}(A_i)=overline{co}left(bigcap_{i}A_iright)$ (i.e., replacing $overline{A_i}$ with $A_i$ in the right side) is false. As a counterexample, let $A_i=(0,2^{-i})cup{1}$ for all integers $ige 1$. Then the left side is $[0,1]$ and the right side is ${1}$.
general-topology vector-spaces convex-hulls
asked Dec 21 '18 at 0:23
Paolo LeonettiPaolo Leonetti
11.5k21550
$endgroup$
add a comment |
$begingroup$
Let $(A_i: i in I)$ be a family of sets in a topological vector space such that for all $i,j in I$ there exists $k in I$ for which $A_i cap A_j=A_k$. Denote by $overline{co}(X)$ the closure of the convex hull of $X$. Is it true that
$$
bigcap_{i in I}overline{co}(A_i)=overline{co}left(bigcap_{i in I}overline{A_i}right),,,?
$$
One inclusion: Since $overline{A_i} subseteq overline{co}(A_i)$ for all $i$ then $bigcap_i overline{A_i} subseteq bigcap_i overline{co}(A_i)$. Therefore
$$
overline{co}left(bigcap_i overline{A_i}right) subseteq overline{co}left(bigcap_i overline{co}(A_i)right)=bigcap_i overline{co}(A_i).
$$
What about the other?
Comment 1. The hypothesis on the family ${A_i:i in I}$ is necessary, see Kavi's answer.
Comment 2. The claim is easily seen to be verified if ${A_i: i in I}$ has a minimum, which holds, e.g., if $I$ is finite.
Comment 3. The claim $bigcap_{i}overline{co}(A_i)=overline{co}left(bigcap_{i}A_iright)$ (i.e., replacing $overline{A_i}$ with $A_i$ in the right side) is false. As a counterexample, let $A_i=(0,2^{-i})cup{1}$ for all integers $ige 1$. Then the left side is $[0,1]$ and the right side is ${1}$.
general-topology vector-spaces convex-hulls
asked Dec 21 '18 at 0:23
Paolo LeonettiPaolo Leonetti
11.5k21550
$endgroup$
Let $(A_i: i in I)$ be a family of sets in a topological vector space such that for all $i,j in I$ there exists $k in I$ for which $A_i cap A_j=A_k$. Denote by $overline{co}(X)$ the closure of the convex hull of $X$. Is it true that
$$
bigcap_{i in I}overline{co}(A_i)=overline{co}left(bigcap_{i in I}overline{A_i}right),,,?
$$
One inclusion: Since $overline{A_i} subseteq overline{co}(A_i)$ for all $i$ then $bigcap_i overline{A_i} subseteq bigcap_i overline{co}(A_i)$. Therefore
$$
overline{co}left(bigcap_i overline{A_i}right) subseteq overline{co}left(bigcap_i overline{co}(A_i)right)=bigcap_i overline{co}(A_i).
$$
What about the other?
Comment 1. The hypothesis on the family ${A_i:i in I}$ is necessary, see Kavi's answer.
Comment 2. The claim is easily seen to be verified if ${A_i: i in I}$ has a minimum, which holds, e.g., if $I$ is finite.
Comment 3. The claim $bigcap_{i}overline{co}(A_i)=overline{co}left(bigcap_{i}A_iright)$ (i.e., replacing $overline{A_i}$ with $A_i$ in the right side) is false. As a counterexample, let $A_i=(0,2^{-i})cup{1}$ for all integers $ige 1$. Then the left side is $[0,1]$ and the right side is ${1}$.
general-topology vector-spaces convex-hulls
general-topology vector-spaces convex-hulls
asked Dec 21 '18 at 0:23
Paolo LeonettiPaolo Leonetti
11.5k21550
asked Dec 21 '18 at 0:23
Paolo LeonettiPaolo Leonetti
11.5k21550
edited Dec 21 '18 at 10:44
Paolo Leonetti
asked Dec 21 '18 at 0:23
Paolo LeonettiPaolo Leonetti
11.5k21550
asked Dec 21 '18 at 0:23
Paolo LeonettiPaolo Leonetti
11.5k21550
asked Dec 21 '18 at 0:23
Paolo LeonettiPaolo Leonetti
11.5k21550
11.5k21550
add a comment |
add a comment |
2 Answers
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$begingroup$
This answer is due to Salvo Tringali.
The claim is false. For each positive integer $n$ let $A_n$ be the set of integers with absolute value greater than $n$. Then the left hand side is $mathbf{R}$ and the right hand side is $emptyset$.
answered Dec 21 '18 at 23:01
Paolo LeonettiPaolo Leonetti
11.5k21550
$endgroup$
add a comment |
$begingroup$
This answer is for an earlier version of the question.
Let $A={1,4}, B={2,3}$. The RHS is empty. But $co(A)=[1,4],co(B)=[2,3]$ so $2 in LHS$.
answered Dec 21 '18 at 0:34
Kavi Rama MurthyKavi Rama Murthy
69.9k53170
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
This answer is due to Salvo Tringali.
The claim is false. For each positive integer $n$ let $A_n$ be the set of integers with absolute value greater than $n$. Then the left hand side is $mathbf{R}$ and the right hand side is $emptyset$.
answered Dec 21 '18 at 23:01
Paolo LeonettiPaolo Leonetti
11.5k21550
$endgroup$
add a comment |
$begingroup$
This answer is due to Salvo Tringali.
The claim is false. For each positive integer $n$ let $A_n$ be the set of integers with absolute value greater than $n$. Then the left hand side is $mathbf{R}$ and the right hand side is $emptyset$.
answered Dec 21 '18 at 23:01
Paolo LeonettiPaolo Leonetti
11.5k21550
$endgroup$
add a comment |
$begingroup$
This answer is due to Salvo Tringali.
The claim is false. For each positive integer $n$ let $A_n$ be the set of integers with absolute value greater than $n$. Then the left hand side is $mathbf{R}$ and the right hand side is $emptyset$.
answered Dec 21 '18 at 23:01
Paolo LeonettiPaolo Leonetti
11.5k21550
$endgroup$
This answer is due to Salvo Tringali.
The claim is false. For each positive integer $n$ let $A_n$ be the set of integers with absolute value greater than $n$. Then the left hand side is $mathbf{R}$ and the right hand side is $emptyset$.
answered Dec 21 '18 at 23:01
Paolo LeonettiPaolo Leonetti
11.5k21550
answered Dec 21 '18 at 23:01
Paolo LeonettiPaolo Leonetti
11.5k21550
answered Dec 21 '18 at 23:01
Paolo LeonettiPaolo Leonetti
11.5k21550
answered Dec 21 '18 at 23:01
Paolo LeonettiPaolo Leonetti
11.5k21550
11.5k21550
add a comment |
add a comment |
$begingroup$
This answer is for an earlier version of the question.
Let $A={1,4}, B={2,3}$. The RHS is empty. But $co(A)=[1,4],co(B)=[2,3]$ so $2 in LHS$.
answered Dec 21 '18 at 0:34
Kavi Rama MurthyKavi Rama Murthy
69.9k53170
$endgroup$
add a comment |
$begingroup$
This answer is for an earlier version of the question.
Let $A={1,4}, B={2,3}$. The RHS is empty. But $co(A)=[1,4],co(B)=[2,3]$ so $2 in LHS$.
answered Dec 21 '18 at 0:34
Kavi Rama MurthyKavi Rama Murthy
69.9k53170
$endgroup$
add a comment |
$begingroup$
This answer is for an earlier version of the question.
Let $A={1,4}, B={2,3}$. The RHS is empty. But $co(A)=[1,4],co(B)=[2,3]$ so $2 in LHS$.
answered Dec 21 '18 at 0:34
Kavi Rama MurthyKavi Rama Murthy
69.9k53170
$endgroup$
This answer is for an earlier version of the question.
Let $A={1,4}, B={2,3}$. The RHS is empty. But $co(A)=[1,4],co(B)=[2,3]$ so $2 in LHS$.
answered Dec 21 '18 at 0:34
Kavi Rama MurthyKavi Rama Murthy
69.9k53170
edited Dec 21 '18 at 0:37
answered Dec 21 '18 at 0:34
Kavi Rama MurthyKavi Rama Murthy
69.9k53170
answered Dec 21 '18 at 0:34
Kavi Rama MurthyKavi Rama Murthy
69.9k53170
answered Dec 21 '18 at 0:34
Kavi Rama MurthyKavi Rama Murthy
69.9k53170
69.9k53170
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