Show that the union of convex sets does not have to be convex.
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The following is an example that I've come up with:
Suppose that $pin A$ and $qin B$ so that $p,q in Acup B$, where $A$ and $B$ are two mutually disjoint, convex, unit circles centered at $x=0,2$ in $mathbb{R^2}$, respectively. Also let $p:=(frac{1}{2},0)$ and $q:= (frac{3}{2},0)$. The set of points satisfying $lambda p + (1-lambda)q$ for $0 < lambda < 1$ forms a line between $p$ and $q$. But for $lambda = frac{1}{2}$, we have that $z = frac{1}{2}p + (1-frac{1}{2})q = frac{1}{2}(p+q)=(1,0)$, which is not in $Acup B$.
I was wondering if there's a simpler example that shows that the union of two convex sets does not have to be convex?
analysis convex-analysis
edited Dec 8 '18 at 14:28
Rodrigo de Azevedo
12.9k41857
asked Dec 8 '18 at 0:09
K.MK.M
693412
$endgroup$
add a comment |
$begingroup$
The following is an example that I've come up with:
Suppose that $pin A$ and $qin B$ so that $p,q in Acup B$, where $A$ and $B$ are two mutually disjoint, convex, unit circles centered at $x=0,2$ in $mathbb{R^2}$, respectively. Also let $p:=(frac{1}{2},0)$ and $q:= (frac{3}{2},0)$. The set of points satisfying $lambda p + (1-lambda)q$ for $0 < lambda < 1$ forms a line between $p$ and $q$. But for $lambda = frac{1}{2}$, we have that $z = frac{1}{2}p + (1-frac{1}{2})q = frac{1}{2}(p+q)=(1,0)$, which is not in $Acup B$.
I was wondering if there's a simpler example that shows that the union of two convex sets does not have to be convex?
analysis convex-analysis
edited Dec 8 '18 at 14:28
Rodrigo de Azevedo
12.9k41857
asked Dec 8 '18 at 0:09
K.MK.M
693412
$endgroup$
add a comment |
$begingroup$
The following is an example that I've come up with:
Suppose that $pin A$ and $qin B$ so that $p,q in Acup B$, where $A$ and $B$ are two mutually disjoint, convex, unit circles centered at $x=0,2$ in $mathbb{R^2}$, respectively. Also let $p:=(frac{1}{2},0)$ and $q:= (frac{3}{2},0)$. The set of points satisfying $lambda p + (1-lambda)q$ for $0 < lambda < 1$ forms a line between $p$ and $q$. But for $lambda = frac{1}{2}$, we have that $z = frac{1}{2}p + (1-frac{1}{2})q = frac{1}{2}(p+q)=(1,0)$, which is not in $Acup B$.
I was wondering if there's a simpler example that shows that the union of two convex sets does not have to be convex?
analysis convex-analysis
edited Dec 8 '18 at 14:28
Rodrigo de Azevedo
12.9k41857
asked Dec 8 '18 at 0:09
K.MK.M
693412
$endgroup$
The following is an example that I've come up with:
Suppose that $pin A$ and $qin B$ so that $p,q in Acup B$, where $A$ and $B$ are two mutually disjoint, convex, unit circles centered at $x=0,2$ in $mathbb{R^2}$, respectively. Also let $p:=(frac{1}{2},0)$ and $q:= (frac{3}{2},0)$. The set of points satisfying $lambda p + (1-lambda)q$ for $0 < lambda < 1$ forms a line between $p$ and $q$. But for $lambda = frac{1}{2}$, we have that $z = frac{1}{2}p + (1-frac{1}{2})q = frac{1}{2}(p+q)=(1,0)$, which is not in $Acup B$.
I was wondering if there's a simpler example that shows that the union of two convex sets does not have to be convex?
analysis convex-analysis
analysis convex-analysis
edited Dec 8 '18 at 14:28
Rodrigo de Azevedo
12.9k41857
asked Dec 8 '18 at 0:09
K.MK.M
693412
edited Dec 8 '18 at 14:28
Rodrigo de Azevedo
12.9k41857
asked Dec 8 '18 at 0:09
K.MK.M
693412
edited Dec 8 '18 at 14:28
Rodrigo de Azevedo
12.9k41857
edited Dec 8 '18 at 14:28
Rodrigo de Azevedo
12.9k41857
edited Dec 8 '18 at 14:28
Rodrigo de Azevedo
12.9k41857
12.9k41857
asked Dec 8 '18 at 0:09
K.MK.M
693412
asked Dec 8 '18 at 0:09
K.MK.M
693412
asked Dec 8 '18 at 0:09
K.MK.M
693412
693412
add a comment |
add a comment |
2 Answers
2
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oldest
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$(0,1) cup (2,3)$ is a simpler example. $frac {0.5+2.5} 2$ does not belong to this union.
answered Dec 8 '18 at 0:13
Kavi Rama MurthyKavi Rama Murthy
57k42159
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add a comment |
$begingroup$
Even easier: two points in the plane.
answered Dec 8 '18 at 0:15
ncmathsadistncmathsadist
42.7k260103
$endgroup$
1
$begingroup$
Or in the line.
$endgroup$
– Martin Argerami
Dec 8 '18 at 0:30
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They "puncture" this conjecture oh-so-prettily.
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– ncmathsadist
Dec 8 '18 at 0:38
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when you say two points in the plane, do you mean that each point is a trivial convex set?
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– K.M
Dec 8 '18 at 0:38
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Verily. A point is about as convex as you can get.
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– ncmathsadist
Dec 8 '18 at 0:38
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@ncmathsadist: wouldn't this be considered more of a counterexample?
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– K.M
Dec 8 '18 at 0:42
add a comment |
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2 Answers
2
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2 Answers
2
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$begingroup$
$(0,1) cup (2,3)$ is a simpler example. $frac {0.5+2.5} 2$ does not belong to this union.
answered Dec 8 '18 at 0:13
Kavi Rama MurthyKavi Rama Murthy
57k42159
$endgroup$
add a comment |
$begingroup$
$(0,1) cup (2,3)$ is a simpler example. $frac {0.5+2.5} 2$ does not belong to this union.
answered Dec 8 '18 at 0:13
Kavi Rama MurthyKavi Rama Murthy
57k42159
$endgroup$
add a comment |
$begingroup$
$(0,1) cup (2,3)$ is a simpler example. $frac {0.5+2.5} 2$ does not belong to this union.
answered Dec 8 '18 at 0:13
Kavi Rama MurthyKavi Rama Murthy
57k42159
$endgroup$
$(0,1) cup (2,3)$ is a simpler example. $frac {0.5+2.5} 2$ does not belong to this union.
answered Dec 8 '18 at 0:13
Kavi Rama MurthyKavi Rama Murthy
57k42159
answered Dec 8 '18 at 0:13
Kavi Rama MurthyKavi Rama Murthy
57k42159
answered Dec 8 '18 at 0:13
Kavi Rama MurthyKavi Rama Murthy
57k42159
answered Dec 8 '18 at 0:13
Kavi Rama MurthyKavi Rama Murthy
57k42159
57k42159
add a comment |
add a comment |
$begingroup$
Even easier: two points in the plane.
answered Dec 8 '18 at 0:15
ncmathsadistncmathsadist
42.7k260103
$endgroup$
1
$begingroup$
Or in the line.
$endgroup$
– Martin Argerami
Dec 8 '18 at 0:30
$begingroup$
They "puncture" this conjecture oh-so-prettily.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
when you say two points in the plane, do you mean that each point is a trivial convex set?
$endgroup$
– K.M
Dec 8 '18 at 0:38
$begingroup$
Verily. A point is about as convex as you can get.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
@ncmathsadist: wouldn't this be considered more of a counterexample?
$endgroup$
– K.M
Dec 8 '18 at 0:42
add a comment |
$begingroup$
Even easier: two points in the plane.
answered Dec 8 '18 at 0:15
ncmathsadistncmathsadist
42.7k260103
$endgroup$
1
$begingroup$
Or in the line.
$endgroup$
– Martin Argerami
Dec 8 '18 at 0:30
$begingroup$
They "puncture" this conjecture oh-so-prettily.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
when you say two points in the plane, do you mean that each point is a trivial convex set?
$endgroup$
– K.M
Dec 8 '18 at 0:38
$begingroup$
Verily. A point is about as convex as you can get.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
@ncmathsadist: wouldn't this be considered more of a counterexample?
$endgroup$
– K.M
Dec 8 '18 at 0:42
add a comment |
$begingroup$
Even easier: two points in the plane.
answered Dec 8 '18 at 0:15
ncmathsadistncmathsadist
42.7k260103
$endgroup$
Even easier: two points in the plane.
answered Dec 8 '18 at 0:15
ncmathsadistncmathsadist
42.7k260103
answered Dec 8 '18 at 0:15
ncmathsadistncmathsadist
42.7k260103
answered Dec 8 '18 at 0:15
ncmathsadistncmathsadist
42.7k260103
answered Dec 8 '18 at 0:15
ncmathsadistncmathsadist
42.7k260103
42.7k260103
1
$begingroup$
Or in the line.
$endgroup$
– Martin Argerami
Dec 8 '18 at 0:30
$begingroup$
They "puncture" this conjecture oh-so-prettily.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
when you say two points in the plane, do you mean that each point is a trivial convex set?
$endgroup$
– K.M
Dec 8 '18 at 0:38
$begingroup$
Verily. A point is about as convex as you can get.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
@ncmathsadist: wouldn't this be considered more of a counterexample?
$endgroup$
– K.M
Dec 8 '18 at 0:42
add a comment |
1
$begingroup$
Or in the line.
$endgroup$
– Martin Argerami
Dec 8 '18 at 0:30
$begingroup$
They "puncture" this conjecture oh-so-prettily.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
when you say two points in the plane, do you mean that each point is a trivial convex set?
$endgroup$
– K.M
Dec 8 '18 at 0:38
$begingroup$
Verily. A point is about as convex as you can get.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
@ncmathsadist: wouldn't this be considered more of a counterexample?
$endgroup$
– K.M
Dec 8 '18 at 0:42
1
1
$begingroup$
Or in the line.
$endgroup$
– Martin Argerami
Dec 8 '18 at 0:30
$begingroup$
Or in the line.
$endgroup$
– Martin Argerami
Dec 8 '18 at 0:30
$begingroup$
They "puncture" this conjecture oh-so-prettily.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
They "puncture" this conjecture oh-so-prettily.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
when you say two points in the plane, do you mean that each point is a trivial convex set?
$endgroup$
– K.M
Dec 8 '18 at 0:38
$begingroup$
when you say two points in the plane, do you mean that each point is a trivial convex set?
$endgroup$
– K.M
Dec 8 '18 at 0:38
$begingroup$
Verily. A point is about as convex as you can get.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
Verily. A point is about as convex as you can get.
$endgroup$
– ncmathsadist
Dec 8 '18 at 0:38
$begingroup$
@ncmathsadist: wouldn't this be considered more of a counterexample?
$endgroup$
– K.M
Dec 8 '18 at 0:42
$begingroup$
@ncmathsadist: wouldn't this be considered more of a counterexample?
$endgroup$
– K.M
Dec 8 '18 at 0:42
add a comment |
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