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







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  • $begingroup$
    a convex ball minus some points at the boundary
    $endgroup$
    – LinAlg
    Dec 1 '18 at 14:16












  • $begingroup$
    @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.
    $endgroup$
    – MPW
    Dec 1 '18 at 14:36






  • 1




    $begingroup$
    @MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
    $endgroup$
    – Andreas Blass
    Dec 1 '18 at 15:11
















0












$begingroup$


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







share|cite|improve this question









$endgroup$












  • $begingroup$
    a convex ball minus some points at the boundary
    $endgroup$
    – LinAlg
    Dec 1 '18 at 14:16












  • $begingroup$
    @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.
    $endgroup$
    – MPW
    Dec 1 '18 at 14:36






  • 1




    $begingroup$
    @MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
    $endgroup$
    – Andreas Blass
    Dec 1 '18 at 15:11














0












0








0





$begingroup$


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







share|cite|improve this question









$endgroup$




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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 1 '18 at 14:10









CrabManCrabMan

199111




199111












  • $begingroup$
    a convex ball minus some points at the boundary
    $endgroup$
    – LinAlg
    Dec 1 '18 at 14:16












  • $begingroup$
    @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.
    $endgroup$
    – MPW
    Dec 1 '18 at 14:36






  • 1




    $begingroup$
    @MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
    $endgroup$
    – Andreas Blass
    Dec 1 '18 at 15:11


















  • $begingroup$
    a convex ball minus some points at the boundary
    $endgroup$
    – LinAlg
    Dec 1 '18 at 14:16












  • $begingroup$
    @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.
    $endgroup$
    – MPW
    Dec 1 '18 at 14:36






  • 1




    $begingroup$
    @MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
    $endgroup$
    – Andreas Blass
    Dec 1 '18 at 15:11
















$begingroup$
a convex ball minus some points at the boundary
$endgroup$
– LinAlg
Dec 1 '18 at 14:16






$begingroup$
a convex ball minus some points at the boundary
$endgroup$
– LinAlg
Dec 1 '18 at 14:16














$begingroup$
@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.
$endgroup$
– MPW
Dec 1 '18 at 14:36




$begingroup$
@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.
$endgroup$
– MPW
Dec 1 '18 at 14:36




1




1




$begingroup$
@MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
$endgroup$
– Andreas Blass
Dec 1 '18 at 15:11




$begingroup$
@MPW I think LinAlg meant "ball" in the sense of Euclidean geometry, i.e., a round ball.
$endgroup$
– Andreas Blass
Dec 1 '18 at 15:11










1 Answer
1






active

oldest

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0












$begingroup$

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.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    "Defined as..."?
    $endgroup$
    – DonAntonio
    Dec 1 '18 at 14:12










  • $begingroup$
    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.
    $endgroup$
    – CrabMan
    Dec 1 '18 at 14:14






  • 1




    $begingroup$
    For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
    $endgroup$
    – Jay
    Dec 1 '18 at 14:29






  • 2




    $begingroup$
    But not really surprising as an intersection of convex sets is always convex, right?
    $endgroup$
    – MPW
    Dec 1 '18 at 14:32











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0












$begingroup$

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.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    "Defined as..."?
    $endgroup$
    – DonAntonio
    Dec 1 '18 at 14:12










  • $begingroup$
    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.
    $endgroup$
    – CrabMan
    Dec 1 '18 at 14:14






  • 1




    $begingroup$
    For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
    $endgroup$
    – Jay
    Dec 1 '18 at 14:29






  • 2




    $begingroup$
    But not really surprising as an intersection of convex sets is always convex, right?
    $endgroup$
    – MPW
    Dec 1 '18 at 14:32
















0












$begingroup$

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.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    "Defined as..."?
    $endgroup$
    – DonAntonio
    Dec 1 '18 at 14:12










  • $begingroup$
    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.
    $endgroup$
    – CrabMan
    Dec 1 '18 at 14:14






  • 1




    $begingroup$
    For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
    $endgroup$
    – Jay
    Dec 1 '18 at 14:29






  • 2




    $begingroup$
    But not really surprising as an intersection of convex sets is always convex, right?
    $endgroup$
    – MPW
    Dec 1 '18 at 14:32














0












0








0





$begingroup$

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.






share|cite|improve this answer











$endgroup$



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.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 1 '18 at 14:13

























answered Dec 1 '18 at 14:10









CrabManCrabMan

199111




199111








  • 1




    $begingroup$
    "Defined as..."?
    $endgroup$
    – DonAntonio
    Dec 1 '18 at 14:12










  • $begingroup$
    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.
    $endgroup$
    – CrabMan
    Dec 1 '18 at 14:14






  • 1




    $begingroup$
    For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
    $endgroup$
    – Jay
    Dec 1 '18 at 14:29






  • 2




    $begingroup$
    But not really surprising as an intersection of convex sets is always convex, right?
    $endgroup$
    – MPW
    Dec 1 '18 at 14:32














  • 1




    $begingroup$
    "Defined as..."?
    $endgroup$
    – DonAntonio
    Dec 1 '18 at 14:12










  • $begingroup$
    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.
    $endgroup$
    – CrabMan
    Dec 1 '18 at 14:14






  • 1




    $begingroup$
    For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
    $endgroup$
    – Jay
    Dec 1 '18 at 14:29






  • 2




    $begingroup$
    But not really surprising as an intersection of convex sets is always convex, right?
    $endgroup$
    – MPW
    Dec 1 '18 at 14:32








1




1




$begingroup$
"Defined as..."?
$endgroup$
– DonAntonio
Dec 1 '18 at 14:12




$begingroup$
"Defined as..."?
$endgroup$
– DonAntonio
Dec 1 '18 at 14:12












$begingroup$
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.
$endgroup$
– CrabMan
Dec 1 '18 at 14:14




$begingroup$
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.
$endgroup$
– CrabMan
Dec 1 '18 at 14:14




1




1




$begingroup$
For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
$endgroup$
– Jay
Dec 1 '18 at 14:29




$begingroup$
For each point $x$ in the boundary of the ball, use the half space containing with boundary flat is tangent to the ball at $x$.
$endgroup$
– Jay
Dec 1 '18 at 14:29




2




2




$begingroup$
But not really surprising as an intersection of convex sets is always convex, right?
$endgroup$
– MPW
Dec 1 '18 at 14:32




$begingroup$
But not really surprising as an intersection of convex sets is always convex, right?
$endgroup$
– MPW
Dec 1 '18 at 14:32


















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