Show that equality for support function of two compact convex sets implies that two sets are equal.
Let $Ssubseteq mathbb{R}^n$.
The support function of set $S$ is defined as the following
$$
sigma_S(x)=sup_{y in S} x^Ty
$$
where $x in mathbb{R}^n$.
Let $F$ and $G$ be two compact convex sets in $mathbb{R}^n$ such that $
sigma_F(x)=sigma_G(x)$.
Show that $F=G$.
Hint: use appropriate separation theorem.
convex-analysis
add a comment |
Let $Ssubseteq mathbb{R}^n$.
The support function of set $S$ is defined as the following
$$
sigma_S(x)=sup_{y in S} x^Ty
$$
where $x in mathbb{R}^n$.
Let $F$ and $G$ be two compact convex sets in $mathbb{R}^n$ such that $
sigma_F(x)=sigma_G(x)$.
Show that $F=G$.
Hint: use appropriate separation theorem.
convex-analysis
add a comment |
Let $Ssubseteq mathbb{R}^n$.
The support function of set $S$ is defined as the following
$$
sigma_S(x)=sup_{y in S} x^Ty
$$
where $x in mathbb{R}^n$.
Let $F$ and $G$ be two compact convex sets in $mathbb{R}^n$ such that $
sigma_F(x)=sigma_G(x)$.
Show that $F=G$.
Hint: use appropriate separation theorem.
convex-analysis
Let $Ssubseteq mathbb{R}^n$.
The support function of set $S$ is defined as the following
$$
sigma_S(x)=sup_{y in S} x^Ty
$$
where $x in mathbb{R}^n$.
Let $F$ and $G$ be two compact convex sets in $mathbb{R}^n$ such that $
sigma_F(x)=sigma_G(x)$.
Show that $F=G$.
Hint: use appropriate separation theorem.
convex-analysis
convex-analysis
asked Nov 24 at 20:27
Sepide
2808
2808
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
This is a straightforward application of Hahn Banach separation theorem: if there is point $u$ in $F$ which is not in $G$ then (we can separate $u$ from $G$ in the sense) there exists a vector $y$ and a real number $r$ such that $y^{T}x <r<y^{T} u $ for all $xin G$. Hence $sigma_G(y) leq r<y^{T} u leq sigma_F(y)$ so $sigma_F(y)neq sigma_G(y)$.
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%2f3012035%2fshow-that-equality-for-support-function-of-two-compact-convex-sets-implies-that%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
This is a straightforward application of Hahn Banach separation theorem: if there is point $u$ in $F$ which is not in $G$ then (we can separate $u$ from $G$ in the sense) there exists a vector $y$ and a real number $r$ such that $y^{T}x <r<y^{T} u $ for all $xin G$. Hence $sigma_G(y) leq r<y^{T} u leq sigma_F(y)$ so $sigma_F(y)neq sigma_G(y)$.
add a comment |
This is a straightforward application of Hahn Banach separation theorem: if there is point $u$ in $F$ which is not in $G$ then (we can separate $u$ from $G$ in the sense) there exists a vector $y$ and a real number $r$ such that $y^{T}x <r<y^{T} u $ for all $xin G$. Hence $sigma_G(y) leq r<y^{T} u leq sigma_F(y)$ so $sigma_F(y)neq sigma_G(y)$.
add a comment |
This is a straightforward application of Hahn Banach separation theorem: if there is point $u$ in $F$ which is not in $G$ then (we can separate $u$ from $G$ in the sense) there exists a vector $y$ and a real number $r$ such that $y^{T}x <r<y^{T} u $ for all $xin G$. Hence $sigma_G(y) leq r<y^{T} u leq sigma_F(y)$ so $sigma_F(y)neq sigma_G(y)$.
This is a straightforward application of Hahn Banach separation theorem: if there is point $u$ in $F$ which is not in $G$ then (we can separate $u$ from $G$ in the sense) there exists a vector $y$ and a real number $r$ such that $y^{T}x <r<y^{T} u $ for all $xin G$. Hence $sigma_G(y) leq r<y^{T} u leq sigma_F(y)$ so $sigma_F(y)neq sigma_G(y)$.
answered Nov 24 at 23:49
Kavi Rama Murthy
48.5k31854
48.5k31854
add a comment |
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3012035%2fshow-that-equality-for-support-function-of-two-compact-convex-sets-implies-that%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