What are some weird examples of convex sets?
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What are some weird examples of convex sets? Maybe sets which intuitively seem non-convex. Or sets not similar to polyhedra, balls, ellipsoids, etc.
convex-analysis
asked Dec 1 at 14:10
CrabMan
199110
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What are some weird examples of convex sets? Maybe sets which intuitively seem non-convex. Or sets not similar to polyhedra, balls, ellipsoids, etc.
convex-analysis
asked Dec 1 at 14:10
CrabMan
199110
a convex ball minus some points at the boundary
– LinAlg
Dec 1 at 14:16
@LinAlg : Even removing a single point from a closed convex ball may not necessarily produce a convex set. For example, the closed unit square in the plane is a closed ball if you choose the right metric, but removing any non-corner point from the boundary produces a nonconvex set.
– MPW
Dec 1 at 14:36
1
@MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
– Andreas Blass
Dec 1 at 15:11
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What are some weird examples of convex sets? Maybe sets which intuitively seem non-convex. Or sets not similar to polyhedra, balls, ellipsoids, etc.
convex-analysis
asked Dec 1 at 14:10
CrabMan
199110
What are some weird examples of convex sets? Maybe sets which intuitively seem non-convex. Or sets not similar to polyhedra, balls, ellipsoids, etc.
convex-analysis
convex-analysis
asked Dec 1 at 14:10
CrabMan
199110
asked Dec 1 at 14:10
CrabMan
199110
asked Dec 1 at 14:10
CrabMan
199110
asked Dec 1 at 14:10
CrabMan
199110
asked Dec 1 at 14:10
CrabMan
199110
199110
a convex ball minus some points at the boundary
– LinAlg
Dec 1 at 14:16
@LinAlg : Even removing a single point from a closed convex ball may not necessarily produce a convex set. For example, the closed unit square in the plane is a closed ball if you choose the right metric, but removing any non-corner point from the boundary produces a nonconvex set.
– MPW
Dec 1 at 14:36
1
@MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
– Andreas Blass
Dec 1 at 15:11
add a comment |
a convex ball minus some points at the boundary
– LinAlg
Dec 1 at 14:16
@LinAlg : Even removing a single point from a closed convex ball may not necessarily produce a convex set. For example, the closed unit square in the plane is a closed ball if you choose the right metric, but removing any non-corner point from the boundary produces a nonconvex set.
– MPW
Dec 1 at 14:36
1
@MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
– Andreas Blass
Dec 1 at 15:11
a convex ball minus some points at the boundary
– LinAlg
Dec 1 at 14:16
a convex ball minus some points at the boundary
– LinAlg
Dec 1 at 14:16
@LinAlg : Even removing a single point from a closed convex ball may not necessarily produce a convex set. For example, the closed unit square in the plane is a closed ball if you choose the right metric, but removing any non-corner point from the boundary produces a nonconvex set.
– MPW
Dec 1 at 14:36
@LinAlg : Even removing a single point from a closed convex ball may not necessarily produce a convex set. For example, the closed unit square in the plane is a closed ball if you choose the right metric, but removing any non-corner point from the boundary produces a nonconvex set.
– MPW
Dec 1 at 14:36
1
1
@MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
– Andreas Blass
Dec 1 at 15:11
@MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
– Andreas Blass
Dec 1 at 15:11
add a comment |
1 Answer
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One example is intersection of countably infinite set of halfspaces.
Consider the infinite sequence $(H_j)_{j=0}^infty$ of halfspaces in $R^2$ defined as $H_j = { x in mathbb{R}^2 mid x_2 geq jx + b_j } $ with for each $j$ having $ b_j $ chosen in such way that the borders of $H_j$ and $H_{j+1}$ intersect at a point with first coordinate equal to $j+1$.
I would say that intersection of this sequence is a somewhat unusual convex set.
answered Dec 1 at 14:10
CrabMan
199110
1
"Defined as..."?
– DonAntonio
Dec 1 at 14:12
I wonder what an intersecrion of a set of halfspaces, which has cardinality continuum, looks like. And I wonder if ball is an example of such a set.
– CrabMan
Dec 1 at 14:14
1
For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
– Jay
Dec 1 at 14:29
2
But not really surprising as an intersection of convex sets is always convex, right?
– MPW
Dec 1 at 14:32
add a comment |
1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
One example is intersection of countably infinite set of halfspaces.
Consider the infinite sequence $(H_j)_{j=0}^infty$ of halfspaces in $R^2$ defined as $H_j = { x in mathbb{R}^2 mid x_2 geq jx + b_j } $ with for each $j$ having $ b_j $ chosen in such way that the borders of $H_j$ and $H_{j+1}$ intersect at a point with first coordinate equal to $j+1$.
I would say that intersection of this sequence is a somewhat unusual convex set.
answered Dec 1 at 14:10
CrabMan
199110
1
"Defined as..."?
– DonAntonio
Dec 1 at 14:12
I wonder what an intersecrion of a set of halfspaces, which has cardinality continuum, looks like. And I wonder if ball is an example of such a set.
– CrabMan
Dec 1 at 14:14
1
For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
– Jay
Dec 1 at 14:29
2
But not really surprising as an intersection of convex sets is always convex, right?
– MPW
Dec 1 at 14:32
add a comment |
up vote
0
down vote
One example is intersection of countably infinite set of halfspaces.
Consider the infinite sequence $(H_j)_{j=0}^infty$ of halfspaces in $R^2$ defined as $H_j = { x in mathbb{R}^2 mid x_2 geq jx + b_j } $ with for each $j$ having $ b_j $ chosen in such way that the borders of $H_j$ and $H_{j+1}$ intersect at a point with first coordinate equal to $j+1$.
I would say that intersection of this sequence is a somewhat unusual convex set.
answered Dec 1 at 14:10
CrabMan
199110
1
"Defined as..."?
– DonAntonio
Dec 1 at 14:12
I wonder what an intersecrion of a set of halfspaces, which has cardinality continuum, looks like. And I wonder if ball is an example of such a set.
– CrabMan
Dec 1 at 14:14
1
For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
– Jay
Dec 1 at 14:29
2
But not really surprising as an intersection of convex sets is always convex, right?
– MPW
Dec 1 at 14:32
add a comment |
up vote
0
down vote
up vote
0
down vote
One example is intersection of countably infinite set of halfspaces.
Consider the infinite sequence $(H_j)_{j=0}^infty$ of halfspaces in $R^2$ defined as $H_j = { x in mathbb{R}^2 mid x_2 geq jx + b_j } $ with for each $j$ having $ b_j $ chosen in such way that the borders of $H_j$ and $H_{j+1}$ intersect at a point with first coordinate equal to $j+1$.
I would say that intersection of this sequence is a somewhat unusual convex set.
answered Dec 1 at 14:10
CrabMan
199110
One example is intersection of countably infinite set of halfspaces.
Consider the infinite sequence $(H_j)_{j=0}^infty$ of halfspaces in $R^2$ defined as $H_j = { x in mathbb{R}^2 mid x_2 geq jx + b_j } $ with for each $j$ having $ b_j $ chosen in such way that the borders of $H_j$ and $H_{j+1}$ intersect at a point with first coordinate equal to $j+1$.
I would say that intersection of this sequence is a somewhat unusual convex set.
answered Dec 1 at 14:10
CrabMan
199110
edited Dec 1 at 14:13
answered Dec 1 at 14:10
CrabMan
199110
answered Dec 1 at 14:10
CrabMan
199110
answered Dec 1 at 14:10
CrabMan
199110
199110
1
"Defined as..."?
– DonAntonio
Dec 1 at 14:12
I wonder what an intersecrion of a set of halfspaces, which has cardinality continuum, looks like. And I wonder if ball is an example of such a set.
– CrabMan
Dec 1 at 14:14
1
For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
– Jay
Dec 1 at 14:29
2
But not really surprising as an intersection of convex sets is always convex, right?
– MPW
Dec 1 at 14:32
add a comment |
1
"Defined as..."?
– DonAntonio
Dec 1 at 14:12
I wonder what an intersecrion of a set of halfspaces, which has cardinality continuum, looks like. And I wonder if ball is an example of such a set.
– CrabMan
Dec 1 at 14:14
1
For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
– Jay
Dec 1 at 14:29
2
But not really surprising as an intersection of convex sets is always convex, right?
– MPW
Dec 1 at 14:32
1
1
"Defined as..."?
– DonAntonio
Dec 1 at 14:12
"Defined as..."?
– DonAntonio
Dec 1 at 14:12
I wonder what an intersecrion of a set of halfspaces, which has cardinality continuum, looks like. And I wonder if ball is an example of such a set.
– CrabMan
Dec 1 at 14:14
I wonder what an intersecrion of a set of halfspaces, which has cardinality continuum, looks like. And I wonder if ball is an example of such a set.
– CrabMan
Dec 1 at 14:14
1
1
For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
– Jay
Dec 1 at 14:29
For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
– Jay
Dec 1 at 14:29
2
2
But not really surprising as an intersection of convex sets is always convex, right?
– MPW
Dec 1 at 14:32
But not really surprising as an intersection of convex sets is always convex, right?
– MPW
Dec 1 at 14:32
add a comment |
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a convex ball minus some points at the boundary
– LinAlg
Dec 1 at 14:16
@LinAlg : Even removing a single point from a closed convex ball may not necessarily produce a convex set. For example, the closed unit square in the plane is a closed ball if you choose the right metric, but removing any non-corner point from the boundary produces a nonconvex set.
– MPW
Dec 1 at 14:36
1
@MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
– Andreas Blass
Dec 1 at 15:11