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 $mathbbR^2$, respectively. Also let $p:=(frac12,0)$ and $q:= (frac32,0)$. The set of points satisfying $lambda p + (1-lambda)q$ for $0 < lambda < 1$ forms a line between $p$ and $q$. But for $lambda = frac12$, we have that $z = frac12p + (1-frac12)q = frac12(p+q)=(1,0)$, 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 at 14:28
Rodrigo de Azevedo
12.8k41853
asked Dec 8 at 0:09
K.M
651312
add a comment |
up vote
2
down vote
favorite
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 $mathbbR^2$, respectively. Also let $p:=(frac12,0)$ and $q:= (frac32,0)$. The set of points satisfying $lambda p + (1-lambda)q$ for $0 < lambda < 1$ forms a line between $p$ and $q$. But for $lambda = frac12$, we have that $z = frac12p + (1-frac12)q = frac12(p+q)=(1,0)$, 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 at 14:28
Rodrigo de Azevedo
12.8k41853
asked Dec 8 at 0:09
K.M
651312
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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 $mathbbR^2$, respectively. Also let $p:=(frac12,0)$ and $q:= (frac32,0)$. The set of points satisfying $lambda p + (1-lambda)q$ for $0 < lambda < 1$ forms a line between $p$ and $q$. But for $lambda = frac12$, we have that $z = frac12p + (1-frac12)q = frac12(p+q)=(1,0)$, 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 at 14:28
Rodrigo de Azevedo
12.8k41853
asked Dec 8 at 0:09
K.M
651312
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 $mathbbR^2$, respectively. Also let $p:=(frac12,0)$ and $q:= (frac32,0)$. The set of points satisfying $lambda p + (1-lambda)q$ for $0 < lambda < 1$ forms a line between $p$ and $q$. But for $lambda = frac12$, we have that $z = frac12p + (1-frac12)q = frac12(p+q)=(1,0)$, 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 at 14:28
Rodrigo de Azevedo
12.8k41853
asked Dec 8 at 0:09
K.M
651312
edited Dec 8 at 14:28
Rodrigo de Azevedo
12.8k41853
asked Dec 8 at 0:09
K.M
651312
edited Dec 8 at 14:28
Rodrigo de Azevedo
12.8k41853
edited Dec 8 at 14:28
Rodrigo de Azevedo
12.8k41853
edited Dec 8 at 14:28
Rodrigo de Azevedo
12.8k41853
12.8k41853
asked Dec 8 at 0:09
K.M
651312
asked Dec 8 at 0:09
K.M
651312
asked Dec 8 at 0:09
K.M
651312
651312
add a comment |
add a comment |
2 Answers
2
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oldest
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up vote
7
down vote
accepted
$(0,1) cup (2,3)$ is a simpler example. $frac 0.5+2.5 2$ does not belong to this union.
answered Dec 8 at 0:13
Kavi Rama Murthy
47.4k31854
add a comment |
up vote
11
down vote
Even easier: two points in the plane.
answered Dec 8 at 0:15
ncmathsadist
42.2k259102
1
Or in the line.
– Martin Argerami
Dec 8 at 0:30
They "puncture" this conjecture oh-so-prettily.
– ncmathsadist
Dec 8 at 0:38
when you say two points in the plane, do you mean that each point is a trivial convex set?
– K.M
Dec 8 at 0:38
Verily. A point is about as convex as you can get.
– ncmathsadist
Dec 8 at 0:38
@ncmathsadist: wouldn't this be considered more of a counterexample?
– K.M
Dec 8 at 0:42
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
accepted
$(0,1) cup (2,3)$ is a simpler example. $frac 0.5+2.5 2$ does not belong to this union.
answered Dec 8 at 0:13
Kavi Rama Murthy
47.4k31854
add a comment |
up vote
7
down vote
accepted
$(0,1) cup (2,3)$ is a simpler example. $frac 0.5+2.5 2$ does not belong to this union.
answered Dec 8 at 0:13
Kavi Rama Murthy
47.4k31854
add a comment |
up vote
7
down vote
accepted
up vote
7
down vote
accepted
$(0,1) cup (2,3)$ is a simpler example. $frac 0.5+2.5 2$ does not belong to this union.
answered Dec 8 at 0:13
Kavi Rama Murthy
47.4k31854
$(0,1) cup (2,3)$ is a simpler example. $frac 0.5+2.5 2$ does not belong to this union.
answered Dec 8 at 0:13
Kavi Rama Murthy
47.4k31854
answered Dec 8 at 0:13
Kavi Rama Murthy
47.4k31854
answered Dec 8 at 0:13
Kavi Rama Murthy
47.4k31854
answered Dec 8 at 0:13
Kavi Rama Murthy
47.4k31854
47.4k31854
add a comment |
add a comment |
up vote
11
down vote
Even easier: two points in the plane.
answered Dec 8 at 0:15
ncmathsadist
42.2k259102
1
Or in the line.
– Martin Argerami
Dec 8 at 0:30
They "puncture" this conjecture oh-so-prettily.
– ncmathsadist
Dec 8 at 0:38
when you say two points in the plane, do you mean that each point is a trivial convex set?
– K.M
Dec 8 at 0:38
Verily. A point is about as convex as you can get.
– ncmathsadist
Dec 8 at 0:38
@ncmathsadist: wouldn't this be considered more of a counterexample?
– K.M
Dec 8 at 0:42
add a comment |
up vote
11
down vote
Even easier: two points in the plane.
answered Dec 8 at 0:15
ncmathsadist
42.2k259102
1
Or in the line.
– Martin Argerami
Dec 8 at 0:30
They "puncture" this conjecture oh-so-prettily.
– ncmathsadist
Dec 8 at 0:38
when you say two points in the plane, do you mean that each point is a trivial convex set?
– K.M
Dec 8 at 0:38
Verily. A point is about as convex as you can get.
– ncmathsadist
Dec 8 at 0:38
@ncmathsadist: wouldn't this be considered more of a counterexample?
– K.M
Dec 8 at 0:42
add a comment |
up vote
11
down vote
up vote
11
down vote
Even easier: two points in the plane.
answered Dec 8 at 0:15
ncmathsadist
42.2k259102
Even easier: two points in the plane.
answered Dec 8 at 0:15
ncmathsadist
42.2k259102
answered Dec 8 at 0:15
ncmathsadist
42.2k259102
answered Dec 8 at 0:15
ncmathsadist
42.2k259102
answered Dec 8 at 0:15
ncmathsadist
42.2k259102
42.2k259102
1
Or in the line.
– Martin Argerami
Dec 8 at 0:30
They "puncture" this conjecture oh-so-prettily.
– ncmathsadist
Dec 8 at 0:38
when you say two points in the plane, do you mean that each point is a trivial convex set?
– K.M
Dec 8 at 0:38
Verily. A point is about as convex as you can get.
– ncmathsadist
Dec 8 at 0:38
@ncmathsadist: wouldn't this be considered more of a counterexample?
– K.M
Dec 8 at 0:42
add a comment |
1
Or in the line.
– Martin Argerami
Dec 8 at 0:30
They "puncture" this conjecture oh-so-prettily.
– ncmathsadist
Dec 8 at 0:38
when you say two points in the plane, do you mean that each point is a trivial convex set?
– K.M
Dec 8 at 0:38
Verily. A point is about as convex as you can get.
– ncmathsadist
Dec 8 at 0:38
@ncmathsadist: wouldn't this be considered more of a counterexample?
– K.M
Dec 8 at 0:42
1
1
Or in the line.
– Martin Argerami
Dec 8 at 0:30
Or in the line.
– Martin Argerami
Dec 8 at 0:30
They "puncture" this conjecture oh-so-prettily.
– ncmathsadist
Dec 8 at 0:38
They "puncture" this conjecture oh-so-prettily.
– ncmathsadist
Dec 8 at 0:38
when you say two points in the plane, do you mean that each point is a trivial convex set?
– K.M
Dec 8 at 0:38
when you say two points in the plane, do you mean that each point is a trivial convex set?
– K.M
Dec 8 at 0:38
Verily. A point is about as convex as you can get.
– ncmathsadist
Dec 8 at 0:38
Verily. A point is about as convex as you can get.
– ncmathsadist
Dec 8 at 0:38
@ncmathsadist: wouldn't this be considered more of a counterexample?
– K.M
Dec 8 at 0:42
@ncmathsadist: wouldn't this be considered more of a counterexample?
– K.M
Dec 8 at 0:42
add a comment |
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