Mixing
how to draw the space of such linear combinations?
Clash Royale CLAN TAG#URR8PPP
up vote
3
down vote
favorite
We have the linear combination
$$ 2 choose 1 x_1 + 1 choose 2 x_2 + 1 choose -2 x_3 + 1 choose 1 x_4 + -1 choose 0 x_5 + 0 choose -1 x_6 $$
As $x_i geq 0 $ is given, according to the definition, the linear combination above generates a cone. But, how can we draw it? Isnt the linear combination above just the entire $mathbbR^2$ plane?
Also, I need to decide whether the vector $6 choose 4$ lies in the cone, but since the linear combination above represesnt the entire plane, then $6 choose 4$ must lie in.
Am I missing something here?
analysis vector-spaces intuition visualization convex-geometry
edited Sep 4 at 7:21
BAYMAX
2,55721021
asked Sep 4 at 6:43
Jimmy Sabater
83512
add a comment |Â
up vote
3
down vote
favorite
We have the linear combination
$$ 2 choose 1 x_1 + 1 choose 2 x_2 + 1 choose -2 x_3 + 1 choose 1 x_4 + -1 choose 0 x_5 + 0 choose -1 x_6 $$
As $x_i geq 0 $ is given, according to the definition, the linear combination above generates a cone. But, how can we draw it? Isnt the linear combination above just the entire $mathbbR^2$ plane?
Also, I need to decide whether the vector $6 choose 4$ lies in the cone, but since the linear combination above represesnt the entire plane, then $6 choose 4$ must lie in.
Am I missing something here?
analysis vector-spaces intuition visualization convex-geometry
edited Sep 4 at 7:21
BAYMAX
2,55721021
asked Sep 4 at 6:43
Jimmy Sabater
83512
can you give reference to your definition?
– BAYMAX
Sep 4 at 6:45
math.stackexchange.com/questions/2904483/…
– Jimmy Sabater
Sep 4 at 6:48
$ 2 choose 1 cdot 4 + 1 choose 2cdot 0+ 1 choose -2 cdot 0 + 1 choose 1cdot 0 + -1 choose 0 cdot 2+ 0 choose -1 cdot 0 = 6 choose 4 $.
– YukiJ
Sep 4 at 6:56
I thought adding the tags of Convex geometry, intuition, visualization would be a good idea, want to reverse it, do tell me!
– BAYMAX
Sep 4 at 7:25
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
We have the linear combination
$$ 2 choose 1 x_1 + 1 choose 2 x_2 + 1 choose -2 x_3 + 1 choose 1 x_4 + -1 choose 0 x_5 + 0 choose -1 x_6 $$
As $x_i geq 0 $ is given, according to the definition, the linear combination above generates a cone. But, how can we draw it? Isnt the linear combination above just the entire $mathbbR^2$ plane?
Also, I need to decide whether the vector $6 choose 4$ lies in the cone, but since the linear combination above represesnt the entire plane, then $6 choose 4$ must lie in.
Am I missing something here?
analysis vector-spaces intuition visualization convex-geometry
edited Sep 4 at 7:21
BAYMAX
2,55721021
asked Sep 4 at 6:43
Jimmy Sabater
83512
We have the linear combination
$$ 2 choose 1 x_1 + 1 choose 2 x_2 + 1 choose -2 x_3 + 1 choose 1 x_4 + -1 choose 0 x_5 + 0 choose -1 x_6 $$
As $x_i geq 0 $ is given, according to the definition, the linear combination above generates a cone. But, how can we draw it? Isnt the linear combination above just the entire $mathbbR^2$ plane?
Also, I need to decide whether the vector $6 choose 4$ lies in the cone, but since the linear combination above represesnt the entire plane, then $6 choose 4$ must lie in.
Am I missing something here?
analysis vector-spaces intuition visualization convex-geometry
edited Sep 4 at 7:21
BAYMAX
2,55721021
asked Sep 4 at 6:43
Jimmy Sabater
83512
edited Sep 4 at 7:21
BAYMAX
2,55721021
edited Sep 4 at 7:21
BAYMAX
2,55721021
edited Sep 4 at 7:21
BAYMAX
2,55721021
2,55721021
asked Sep 4 at 6:43
Jimmy Sabater
83512
asked Sep 4 at 6:43
Jimmy Sabater
83512
asked Sep 4 at 6:43
Jimmy Sabater
83512
83512
can you give reference to your definition?
– BAYMAX
Sep 4 at 6:45
math.stackexchange.com/questions/2904483/…
– Jimmy Sabater
Sep 4 at 6:48
$ 2 choose 1 cdot 4 + 1 choose 2cdot 0+ 1 choose -2 cdot 0 + 1 choose 1cdot 0 + -1 choose 0 cdot 2+ 0 choose -1 cdot 0 = 6 choose 4 $.
– YukiJ
Sep 4 at 6:56
I thought adding the tags of Convex geometry, intuition, visualization would be a good idea, want to reverse it, do tell me!
– BAYMAX
Sep 4 at 7:25
add a comment |Â
can you give reference to your definition?
– BAYMAX
Sep 4 at 6:45
math.stackexchange.com/questions/2904483/…
– Jimmy Sabater
Sep 4 at 6:48
$ 2 choose 1 cdot 4 + 1 choose 2cdot 0+ 1 choose -2 cdot 0 + 1 choose 1cdot 0 + -1 choose 0 cdot 2+ 0 choose -1 cdot 0 = 6 choose 4 $.
– YukiJ
Sep 4 at 6:56
I thought adding the tags of Convex geometry, intuition, visualization would be a good idea, want to reverse it, do tell me!
– BAYMAX
Sep 4 at 7:25
can you give reference to your definition?
– BAYMAX
Sep 4 at 6:45
can you give reference to your definition?
– BAYMAX
Sep 4 at 6:45
math.stackexchange.com/questions/2904483/…
– Jimmy Sabater
Sep 4 at 6:48
math.stackexchange.com/questions/2904483/…
– Jimmy Sabater
Sep 4 at 6:48
$ 2 choose 1 cdot 4 + 1 choose 2cdot 0+ 1 choose -2 cdot 0 + 1 choose 1cdot 0 + -1 choose 0 cdot 2+ 0 choose -1 cdot 0 = 6 choose 4 $.
– YukiJ
Sep 4 at 6:56
$ 2 choose 1 cdot 4 + 1 choose 2cdot 0+ 1 choose -2 cdot 0 + 1 choose 1cdot 0 + -1 choose 0 cdot 2+ 0 choose -1 cdot 0 = 6 choose 4 $.
– YukiJ
Sep 4 at 6:56
I thought adding the tags of Convex geometry, intuition, visualization would be a good idea, want to reverse it, do tell me!
– BAYMAX
Sep 4 at 7:25
I thought adding the tags of Convex geometry, intuition, visualization would be a good idea, want to reverse it, do tell me!
– BAYMAX
Sep 4 at 7:25
add a comment |Â
4 Answers
4
active
oldest
votes
up vote
7
down vote
accepted
Just to illustrate Yves' answer: in the below figure the original points are drawn in blue. The blue polygon is their convex hull. As Yves rightly pointed out, the origin (black) is inside this convex hull. To draw the cone, you now need to inflate this blue convex hull. This is illustrated by the red polygon. You will notice that you will ultimately end up with the whole plane when you keep on inflating the convex hull. Hence, the cone you are looking for is indeed the whole plane and since $6 choose 4 $ is obviously in the whole plane, it is also in the cone. Hope this helps.
edited Sep 4 at 8:51
Filippo De Bortoli
997521
answered Sep 4 at 7:48
YukiJ
1,6742624
what is a convex hull?
– Jimmy Sabater
Sep 4 at 8:00
2
The convex hull of a set of points is smallest convex set which contains all of the points. A set is called convex if for any two points in the set the line between those two points lies completely in the set. This implies that one property of a convex set is that it has no holes.
– YukiJ
Sep 4 at 8:02
So, to draw the convex hull, we just plot the point and join them to form a polygon? How about the point (1,1)?
– Jimmy Sabater
Sep 4 at 8:05
You have to join them and make sure at the same time that the resulting polygon will be convex. Notice that (1,1) lies inside the interior of the convex hull. If you joined any of the other points to (1,1) and used the line between the two as an edge of the polygon, the resulting polygon would not be convex. For example, if the edges of the polygon were drawn from (-1,0)~(1,1)~(1,2)~(2,1)~(1,-2)~(0,-1)~(-1,0) then this polygon would not be convex because the line between (-1,0) and (1,2) would not lie inside the polygon.
– YukiJ
Sep 4 at 8:11
add a comment |Â
up vote
2
down vote
If $max(u,v)lt0$, then
$$
beginpmatrixu\vendpmatrix=(-u)beginpmatrix-1\0endpmatrix+(-v)beginpmatrix0\-1endpmatrix
$$
If $max(u,v)ge0$, then
$$
beginpmatrixu\vendpmatrix=max(u,v)beginpmatrix1\1endpmatrix+max(v-u,0)beginpmatrix-1\0endpmatrix+max(u-v,0)beginpmatrix0\-1endpmatrix
$$
answered Sep 4 at 7:48
robjohn♦
259k26298613
add a comment |Â
up vote
2
down vote
A possible way is as follows:
- Let's call the vectors $v_1=2 choose 1 , v_2 =1 choose 2, v_3=1 choose -2, v_4 = 1 choose 1, v_5=-1 choose 0 , v_6=0 choose -1 $
- Excluding the trivial case $x_1 = ldots = x_6 = 0$ note that
- $S = sum_i=1^6x_i > 0 Rightarrow sum_i=1^6fracx_iSv_i$ is a convex combination of the given vectors.
- So, the convex hull of the given vectors $v_1, ldots , v_6$ must lie in the cone.
- If this covex hull contains an open ball around $ 0choose 0 $ - which is the case - then the "cone" is the whole plane, because any positive multiple of the elements of this convex hull must lie in the cone, as well.
answered Sep 4 at 6:59
trancelocation
5,2101515
add a comment |Â
up vote
1
down vote
You are right. As the origin lies inside the convex hull of the vectors, any point can be reached.
answered Sep 4 at 6:57
Yves Daoust
113k665208
how can we draw that cone by hand?
– Jimmy Sabater
Sep 4 at 6:58
1
Draw the points, draw the hull and inflate it.
– Yves Daoust
Sep 4 at 6:59
hmm... why the downvote?
– Siong Thye Goh
Sep 4 at 7:33
@SiongThyeGoh: thanks for the solidarity ;-)
– Yves Daoust
Sep 4 at 7:35
1
@FedericoPoloni: my personal policy is to write answers as short as I can (without sacrificing meaning) and to stick to the essential. Some will prefer a more tutorial approach. My motivation is certainly not getting reputation, I already got my tee-shirt ;-)
– Yves Daoust
Sep 4 at 12:23
 |Â
show 3 more comments
4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
7
down vote
accepted
Just to illustrate Yves' answer: in the below figure the original points are drawn in blue. The blue polygon is their convex hull. As Yves rightly pointed out, the origin (black) is inside this convex hull. To draw the cone, you now need to inflate this blue convex hull. This is illustrated by the red polygon. You will notice that you will ultimately end up with the whole plane when you keep on inflating the convex hull. Hence, the cone you are looking for is indeed the whole plane and since $6 choose 4 $ is obviously in the whole plane, it is also in the cone. Hope this helps.
edited Sep 4 at 8:51
Filippo De Bortoli
997521
answered Sep 4 at 7:48
YukiJ
1,6742624
what is a convex hull?
– Jimmy Sabater
Sep 4 at 8:00
2
The convex hull of a set of points is smallest convex set which contains all of the points. A set is called convex if for any two points in the set the line between those two points lies completely in the set. This implies that one property of a convex set is that it has no holes.
– YukiJ
Sep 4 at 8:02
So, to draw the convex hull, we just plot the point and join them to form a polygon? How about the point (1,1)?
– Jimmy Sabater
Sep 4 at 8:05
You have to join them and make sure at the same time that the resulting polygon will be convex. Notice that (1,1) lies inside the interior of the convex hull. If you joined any of the other points to (1,1) and used the line between the two as an edge of the polygon, the resulting polygon would not be convex. For example, if the edges of the polygon were drawn from (-1,0)~(1,1)~(1,2)~(2,1)~(1,-2)~(0,-1)~(-1,0) then this polygon would not be convex because the line between (-1,0) and (1,2) would not lie inside the polygon.
– YukiJ
Sep 4 at 8:11
add a comment |Â
up vote
7
down vote
accepted
Just to illustrate Yves' answer: in the below figure the original points are drawn in blue. The blue polygon is their convex hull. As Yves rightly pointed out, the origin (black) is inside this convex hull. To draw the cone, you now need to inflate this blue convex hull. This is illustrated by the red polygon. You will notice that you will ultimately end up with the whole plane when you keep on inflating the convex hull. Hence, the cone you are looking for is indeed the whole plane and since $6 choose 4 $ is obviously in the whole plane, it is also in the cone. Hope this helps.
edited Sep 4 at 8:51
Filippo De Bortoli
997521
answered Sep 4 at 7:48
YukiJ
1,6742624
what is a convex hull?
– Jimmy Sabater
Sep 4 at 8:00
2
The convex hull of a set of points is smallest convex set which contains all of the points. A set is called convex if for any two points in the set the line between those two points lies completely in the set. This implies that one property of a convex set is that it has no holes.
– YukiJ
Sep 4 at 8:02
So, to draw the convex hull, we just plot the point and join them to form a polygon? How about the point (1,1)?
– Jimmy Sabater
Sep 4 at 8:05
You have to join them and make sure at the same time that the resulting polygon will be convex. Notice that (1,1) lies inside the interior of the convex hull. If you joined any of the other points to (1,1) and used the line between the two as an edge of the polygon, the resulting polygon would not be convex. For example, if the edges of the polygon were drawn from (-1,0)~(1,1)~(1,2)~(2,1)~(1,-2)~(0,-1)~(-1,0) then this polygon would not be convex because the line between (-1,0) and (1,2) would not lie inside the polygon.
– YukiJ
Sep 4 at 8:11
add a comment |Â
up vote
7
down vote
accepted
up vote
7
down vote
accepted
Just to illustrate Yves' answer: in the below figure the original points are drawn in blue. The blue polygon is their convex hull. As Yves rightly pointed out, the origin (black) is inside this convex hull. To draw the cone, you now need to inflate this blue convex hull. This is illustrated by the red polygon. You will notice that you will ultimately end up with the whole plane when you keep on inflating the convex hull. Hence, the cone you are looking for is indeed the whole plane and since $6 choose 4 $ is obviously in the whole plane, it is also in the cone. Hope this helps.
edited Sep 4 at 8:51
Filippo De Bortoli
997521
answered Sep 4 at 7:48
YukiJ
1,6742624
Just to illustrate Yves' answer: in the below figure the original points are drawn in blue. The blue polygon is their convex hull. As Yves rightly pointed out, the origin (black) is inside this convex hull. To draw the cone, you now need to inflate this blue convex hull. This is illustrated by the red polygon. You will notice that you will ultimately end up with the whole plane when you keep on inflating the convex hull. Hence, the cone you are looking for is indeed the whole plane and since $6 choose 4 $ is obviously in the whole plane, it is also in the cone. Hope this helps.
edited Sep 4 at 8:51
Filippo De Bortoli
997521
answered Sep 4 at 7:48
YukiJ
1,6742624
edited Sep 4 at 8:51
Filippo De Bortoli
997521
edited Sep 4 at 8:51
Filippo De Bortoli
997521
edited Sep 4 at 8:51
Filippo De Bortoli
997521
997521
answered Sep 4 at 7:48
YukiJ
1,6742624
answered Sep 4 at 7:48
YukiJ
1,6742624
answered Sep 4 at 7:48
YukiJ
1,6742624
1,6742624
what is a convex hull?
– Jimmy Sabater
Sep 4 at 8:00
2
The convex hull of a set of points is smallest convex set which contains all of the points. A set is called convex if for any two points in the set the line between those two points lies completely in the set. This implies that one property of a convex set is that it has no holes.
– YukiJ
Sep 4 at 8:02
So, to draw the convex hull, we just plot the point and join them to form a polygon? How about the point (1,1)?
– Jimmy Sabater
Sep 4 at 8:05
You have to join them and make sure at the same time that the resulting polygon will be convex. Notice that (1,1) lies inside the interior of the convex hull. If you joined any of the other points to (1,1) and used the line between the two as an edge of the polygon, the resulting polygon would not be convex. For example, if the edges of the polygon were drawn from (-1,0)~(1,1)~(1,2)~(2,1)~(1,-2)~(0,-1)~(-1,0) then this polygon would not be convex because the line between (-1,0) and (1,2) would not lie inside the polygon.
– YukiJ
Sep 4 at 8:11
add a comment |Â
what is a convex hull?
– Jimmy Sabater
Sep 4 at 8:00
2
The convex hull of a set of points is smallest convex set which contains all of the points. A set is called convex if for any two points in the set the line between those two points lies completely in the set. This implies that one property of a convex set is that it has no holes.
– YukiJ
Sep 4 at 8:02
So, to draw the convex hull, we just plot the point and join them to form a polygon? How about the point (1,1)?
– Jimmy Sabater
Sep 4 at 8:05
You have to join them and make sure at the same time that the resulting polygon will be convex. Notice that (1,1) lies inside the interior of the convex hull. If you joined any of the other points to (1,1) and used the line between the two as an edge of the polygon, the resulting polygon would not be convex. For example, if the edges of the polygon were drawn from (-1,0)~(1,1)~(1,2)~(2,1)~(1,-2)~(0,-1)~(-1,0) then this polygon would not be convex because the line between (-1,0) and (1,2) would not lie inside the polygon.
– YukiJ
Sep 4 at 8:11
what is a convex hull?
– Jimmy Sabater
Sep 4 at 8:00
what is a convex hull?
– Jimmy Sabater
Sep 4 at 8:00
2
2
The convex hull of a set of points is smallest convex set which contains all of the points. A set is called convex if for any two points in the set the line between those two points lies completely in the set. This implies that one property of a convex set is that it has no holes.
– YukiJ
Sep 4 at 8:02
The convex hull of a set of points is smallest convex set which contains all of the points. A set is called convex if for any two points in the set the line between those two points lies completely in the set. This implies that one property of a convex set is that it has no holes.
– YukiJ
Sep 4 at 8:02
So, to draw the convex hull, we just plot the point and join them to form a polygon? How about the point (1,1)?
– Jimmy Sabater
Sep 4 at 8:05
So, to draw the convex hull, we just plot the point and join them to form a polygon? How about the point (1,1)?
– Jimmy Sabater
Sep 4 at 8:05
You have to join them and make sure at the same time that the resulting polygon will be convex. Notice that (1,1) lies inside the interior of the convex hull. If you joined any of the other points to (1,1) and used the line between the two as an edge of the polygon, the resulting polygon would not be convex. For example, if the edges of the polygon were drawn from (-1,0)~(1,1)~(1,2)~(2,1)~(1,-2)~(0,-1)~(-1,0) then this polygon would not be convex because the line between (-1,0) and (1,2) would not lie inside the polygon.
– YukiJ
Sep 4 at 8:11
You have to join them and make sure at the same time that the resulting polygon will be convex. Notice that (1,1) lies inside the interior of the convex hull. If you joined any of the other points to (1,1) and used the line between the two as an edge of the polygon, the resulting polygon would not be convex. For example, if the edges of the polygon were drawn from (-1,0)~(1,1)~(1,2)~(2,1)~(1,-2)~(0,-1)~(-1,0) then this polygon would not be convex because the line between (-1,0) and (1,2) would not lie inside the polygon.
– YukiJ
Sep 4 at 8:11
add a comment |Â
up vote
2
down vote
If $max(u,v)lt0$, then
$$
beginpmatrixu\vendpmatrix=(-u)beginpmatrix-1\0endpmatrix+(-v)beginpmatrix0\-1endpmatrix
$$
If $max(u,v)ge0$, then
$$
beginpmatrixu\vendpmatrix=max(u,v)beginpmatrix1\1endpmatrix+max(v-u,0)beginpmatrix-1\0endpmatrix+max(u-v,0)beginpmatrix0\-1endpmatrix
$$
answered Sep 4 at 7:48
robjohn♦
259k26298613
add a comment |Â
up vote
2
down vote
If $max(u,v)lt0$, then
$$
beginpmatrixu\vendpmatrix=(-u)beginpmatrix-1\0endpmatrix+(-v)beginpmatrix0\-1endpmatrix
$$
If $max(u,v)ge0$, then
$$
beginpmatrixu\vendpmatrix=max(u,v)beginpmatrix1\1endpmatrix+max(v-u,0)beginpmatrix-1\0endpmatrix+max(u-v,0)beginpmatrix0\-1endpmatrix
$$
answered Sep 4 at 7:48
robjohn♦
259k26298613
add a comment |Â
up vote
2
down vote
up vote
2
down vote
If $max(u,v)lt0$, then
$$
beginpmatrixu\vendpmatrix=(-u)beginpmatrix-1\0endpmatrix+(-v)beginpmatrix0\-1endpmatrix
$$
If $max(u,v)ge0$, then
$$
beginpmatrixu\vendpmatrix=max(u,v)beginpmatrix1\1endpmatrix+max(v-u,0)beginpmatrix-1\0endpmatrix+max(u-v,0)beginpmatrix0\-1endpmatrix
$$
answered Sep 4 at 7:48
robjohn♦
259k26298613
If $max(u,v)lt0$, then
$$
beginpmatrixu\vendpmatrix=(-u)beginpmatrix-1\0endpmatrix+(-v)beginpmatrix0\-1endpmatrix
$$
If $max(u,v)ge0$, then
$$
beginpmatrixu\vendpmatrix=max(u,v)beginpmatrix1\1endpmatrix+max(v-u,0)beginpmatrix-1\0endpmatrix+max(u-v,0)beginpmatrix0\-1endpmatrix
$$
answered Sep 4 at 7:48
robjohn♦
259k26298613
edited Sep 4 at 7:57
answered Sep 4 at 7:48
robjohn♦
259k26298613
answered Sep 4 at 7:48
robjohn♦
259k26298613
answered Sep 4 at 7:48
robjohn♦
259k26298613
259k26298613
add a comment |Â
add a comment |Â
up vote
2
down vote
A possible way is as follows:
- Let's call the vectors $v_1=2 choose 1 , v_2 =1 choose 2, v_3=1 choose -2, v_4 = 1 choose 1, v_5=-1 choose 0 , v_6=0 choose -1 $
- Excluding the trivial case $x_1 = ldots = x_6 = 0$ note that
- $S = sum_i=1^6x_i > 0 Rightarrow sum_i=1^6fracx_iSv_i$ is a convex combination of the given vectors.
- So, the convex hull of the given vectors $v_1, ldots , v_6$ must lie in the cone.
- If this covex hull contains an open ball around $ 0choose 0 $ - which is the case - then the "cone" is the whole plane, because any positive multiple of the elements of this convex hull must lie in the cone, as well.
answered Sep 4 at 6:59
trancelocation
5,2101515
add a comment |Â
up vote
2
down vote
A possible way is as follows:
- Let's call the vectors $v_1=2 choose 1 , v_2 =1 choose 2, v_3=1 choose -2, v_4 = 1 choose 1, v_5=-1 choose 0 , v_6=0 choose -1 $
- Excluding the trivial case $x_1 = ldots = x_6 = 0$ note that
- $S = sum_i=1^6x_i > 0 Rightarrow sum_i=1^6fracx_iSv_i$ is a convex combination of the given vectors.
- So, the convex hull of the given vectors $v_1, ldots , v_6$ must lie in the cone.
- If this covex hull contains an open ball around $ 0choose 0 $ - which is the case - then the "cone" is the whole plane, because any positive multiple of the elements of this convex hull must lie in the cone, as well.
answered Sep 4 at 6:59
trancelocation
5,2101515
add a comment |Â
up vote
2
down vote
up vote
2
down vote
A possible way is as follows:
- Let's call the vectors $v_1=2 choose 1 , v_2 =1 choose 2, v_3=1 choose -2, v_4 = 1 choose 1, v_5=-1 choose 0 , v_6=0 choose -1 $
- Excluding the trivial case $x_1 = ldots = x_6 = 0$ note that
- $S = sum_i=1^6x_i > 0 Rightarrow sum_i=1^6fracx_iSv_i$ is a convex combination of the given vectors.
- So, the convex hull of the given vectors $v_1, ldots , v_6$ must lie in the cone.
- If this covex hull contains an open ball around $ 0choose 0 $ - which is the case - then the "cone" is the whole plane, because any positive multiple of the elements of this convex hull must lie in the cone, as well.
answered Sep 4 at 6:59
trancelocation
5,2101515
A possible way is as follows:
- Let's call the vectors $v_1=2 choose 1 , v_2 =1 choose 2, v_3=1 choose -2, v_4 = 1 choose 1, v_5=-1 choose 0 , v_6=0 choose -1 $
- Excluding the trivial case $x_1 = ldots = x_6 = 0$ note that
- $S = sum_i=1^6x_i > 0 Rightarrow sum_i=1^6fracx_iSv_i$ is a convex combination of the given vectors.
- So, the convex hull of the given vectors $v_1, ldots , v_6$ must lie in the cone.
- If this covex hull contains an open ball around $ 0choose 0 $ - which is the case - then the "cone" is the whole plane, because any positive multiple of the elements of this convex hull must lie in the cone, as well.
answered Sep 4 at 6:59
trancelocation
5,2101515
edited Sep 4 at 8:19
answered Sep 4 at 6:59
trancelocation
5,2101515
answered Sep 4 at 6:59
trancelocation
5,2101515
answered Sep 4 at 6:59
trancelocation
5,2101515
5,2101515
add a comment |Â
add a comment |Â
up vote
1
down vote
You are right. As the origin lies inside the convex hull of the vectors, any point can be reached.
answered Sep 4 at 6:57
Yves Daoust
113k665208
how can we draw that cone by hand?
– Jimmy Sabater
Sep 4 at 6:58
1
Draw the points, draw the hull and inflate it.
– Yves Daoust
Sep 4 at 6:59
hmm... why the downvote?
– Siong Thye Goh
Sep 4 at 7:33
@SiongThyeGoh: thanks for the solidarity ;-)
– Yves Daoust
Sep 4 at 7:35
1
@FedericoPoloni: my personal policy is to write answers as short as I can (without sacrificing meaning) and to stick to the essential. Some will prefer a more tutorial approach. My motivation is certainly not getting reputation, I already got my tee-shirt ;-)
– Yves Daoust
Sep 4 at 12:23
 |Â
show 3 more comments
up vote
1
down vote
You are right. As the origin lies inside the convex hull of the vectors, any point can be reached.
answered Sep 4 at 6:57
Yves Daoust
113k665208
how can we draw that cone by hand?
– Jimmy Sabater
Sep 4 at 6:58
1
Draw the points, draw the hull and inflate it.
– Yves Daoust
Sep 4 at 6:59
hmm... why the downvote?
– Siong Thye Goh
Sep 4 at 7:33
@SiongThyeGoh: thanks for the solidarity ;-)
– Yves Daoust
Sep 4 at 7:35
1
@FedericoPoloni: my personal policy is to write answers as short as I can (without sacrificing meaning) and to stick to the essential. Some will prefer a more tutorial approach. My motivation is certainly not getting reputation, I already got my tee-shirt ;-)
– Yves Daoust
Sep 4 at 12:23
 |Â
show 3 more comments
up vote
1
down vote
up vote
1
down vote
You are right. As the origin lies inside the convex hull of the vectors, any point can be reached.
answered Sep 4 at 6:57
Yves Daoust
113k665208
You are right. As the origin lies inside the convex hull of the vectors, any point can be reached.
answered Sep 4 at 6:57
Yves Daoust
113k665208
answered Sep 4 at 6:57
Yves Daoust
113k665208
answered Sep 4 at 6:57
Yves Daoust
113k665208
answered Sep 4 at 6:57
Yves Daoust
113k665208
113k665208
how can we draw that cone by hand?
– Jimmy Sabater
Sep 4 at 6:58
1
Draw the points, draw the hull and inflate it.
– Yves Daoust
Sep 4 at 6:59
hmm... why the downvote?
– Siong Thye Goh
Sep 4 at 7:33
@SiongThyeGoh: thanks for the solidarity ;-)
– Yves Daoust
Sep 4 at 7:35
1
@FedericoPoloni: my personal policy is to write answers as short as I can (without sacrificing meaning) and to stick to the essential. Some will prefer a more tutorial approach. My motivation is certainly not getting reputation, I already got my tee-shirt ;-)
– Yves Daoust
Sep 4 at 12:23
 |Â
show 3 more comments
how can we draw that cone by hand?
– Jimmy Sabater
Sep 4 at 6:58
1
Draw the points, draw the hull and inflate it.
– Yves Daoust
Sep 4 at 6:59
hmm... why the downvote?
– Siong Thye Goh
Sep 4 at 7:33
@SiongThyeGoh: thanks for the solidarity ;-)
– Yves Daoust
Sep 4 at 7:35
1
@FedericoPoloni: my personal policy is to write answers as short as I can (without sacrificing meaning) and to stick to the essential. Some will prefer a more tutorial approach. My motivation is certainly not getting reputation, I already got my tee-shirt ;-)
– Yves Daoust
Sep 4 at 12:23
how can we draw that cone by hand?
– Jimmy Sabater
Sep 4 at 6:58
how can we draw that cone by hand?
– Jimmy Sabater
Sep 4 at 6:58
1
1
Draw the points, draw the hull and inflate it.
– Yves Daoust
Sep 4 at 6:59
Draw the points, draw the hull and inflate it.
– Yves Daoust
Sep 4 at 6:59
hmm... why the downvote?
– Siong Thye Goh
Sep 4 at 7:33
hmm... why the downvote?
– Siong Thye Goh
Sep 4 at 7:33
@SiongThyeGoh: thanks for the solidarity ;-)
– Yves Daoust
Sep 4 at 7:35
@SiongThyeGoh: thanks for the solidarity ;-)
– Yves Daoust
Sep 4 at 7:35
1
1
@FedericoPoloni: my personal policy is to write answers as short as I can (without sacrificing meaning) and to stick to the essential. Some will prefer a more tutorial approach. My motivation is certainly not getting reputation, I already got my tee-shirt ;-)
– Yves Daoust
Sep 4 at 12:23
@FedericoPoloni: my personal policy is to write answers as short as I can (without sacrificing meaning) and to stick to the essential. Some will prefer a more tutorial approach. My motivation is certainly not getting reputation, I already got my tee-shirt ;-)
– Yves Daoust
Sep 4 at 12:23
 |Â
show 3 more comments
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can you give reference to your definition?
– BAYMAX
Sep 4 at 6:45
math.stackexchange.com/questions/2904483/…
– Jimmy Sabater
Sep 4 at 6:48
$ 2 choose 1 cdot 4 + 1 choose 2cdot 0+ 1 choose -2 cdot 0 + 1 choose 1cdot 0 + -1 choose 0 cdot 2+ 0 choose -1 cdot 0 = 6 choose 4 $.
– YukiJ
Sep 4 at 6:56
I thought adding the tags of Convex geometry, intuition, visualization would be a good idea, want to reverse it, do tell me!
– BAYMAX
Sep 4 at 7:25