What are some weird examples of convex sets?
$begingroup$
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
asked Dec 1 '18 at 14:10
CrabManCrabMan
199111
$endgroup$
add a comment |
$begingroup$
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
asked Dec 1 '18 at 14:10
CrabManCrabMan
199111
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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
asked Dec 1 '18 at 14:10
CrabManCrabMan
199111
$endgroup$
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
asked Dec 1 '18 at 14:10
CrabManCrabMan
199111
asked Dec 1 '18 at 14:10
CrabManCrabMan
199111
asked Dec 1 '18 at 14:10
CrabManCrabMan
199111
asked Dec 1 '18 at 14:10
CrabManCrabMan
199111
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
add a comment |
$begingroup$
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
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
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.
answered Dec 1 '18 at 14:10
CrabManCrabMan
199111
$endgroup$
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
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3021386%2fwhat-are-some-weird-examples-of-convex-sets%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
answered Dec 1 '18 at 14:10
CrabManCrabMan
199111
$endgroup$
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
add a comment |
$begingroup$
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.
answered Dec 1 '18 at 14:10
CrabManCrabMan
199111
$endgroup$
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
add a comment |
$begingroup$
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.
answered Dec 1 '18 at 14:10
CrabManCrabMan
199111
$endgroup$
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.
answered Dec 1 '18 at 14:10
CrabManCrabMan
199111
edited Dec 1 '18 at 14:13
answered Dec 1 '18 at 14:10
CrabManCrabMan
199111
answered Dec 1 '18 at 14:10
CrabManCrabMan
199111
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
add a comment |
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
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3021386%2fwhat-are-some-weird-examples-of-convex-sets%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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