Mixing
A natural concave function
Let $mathcal{S}$ denote a compact, convex set, $mathcal{B}$ be it's boundary and $mathcal{D}(.,.)$ a convex symmetric distance defined on it. i.e.
$$mathcal{D}(x,y)=mathcal{D}(y,x)$$
and
$$mathcal{D}(x, lambda y_1+(1-lambda)y_2)leqlambdamathcal{D}(x,y_1)+(1-lambda)mathcal{D}(x,y_2)$$
for all $lambdain[0,1]$ and $x,y,y_1,y_2inmathcal{S}$.
Is the following function concave?
$$f(x):=min_{binmathcal{B}}mathcal{D}(x,b)$$
i.e. can you prove
$$f(lambda x_1+(1-lambda)x_2)geq lambda f(x_1)+(1-lambda)f(x_2)$$
?
convex-analysis
asked Oct 26 '18 at 18:06
K. Sadri
696
add a comment |
Let $mathcal{S}$ denote a compact, convex set, $mathcal{B}$ be it's boundary and $mathcal{D}(.,.)$ a convex symmetric distance defined on it. i.e.
$$mathcal{D}(x,y)=mathcal{D}(y,x)$$
and
$$mathcal{D}(x, lambda y_1+(1-lambda)y_2)leqlambdamathcal{D}(x,y_1)+(1-lambda)mathcal{D}(x,y_2)$$
for all $lambdain[0,1]$ and $x,y,y_1,y_2inmathcal{S}$.
Is the following function concave?
$$f(x):=min_{binmathcal{B}}mathcal{D}(x,b)$$
i.e. can you prove
$$f(lambda x_1+(1-lambda)x_2)geq lambda f(x_1)+(1-lambda)f(x_2)$$
?
convex-analysis
asked Oct 26 '18 at 18:06
K. Sadri
696
add a comment |
Let $mathcal{S}$ denote a compact, convex set, $mathcal{B}$ be it's boundary and $mathcal{D}(.,.)$ a convex symmetric distance defined on it. i.e.
$$mathcal{D}(x,y)=mathcal{D}(y,x)$$
and
$$mathcal{D}(x, lambda y_1+(1-lambda)y_2)leqlambdamathcal{D}(x,y_1)+(1-lambda)mathcal{D}(x,y_2)$$
for all $lambdain[0,1]$ and $x,y,y_1,y_2inmathcal{S}$.
Is the following function concave?
$$f(x):=min_{binmathcal{B}}mathcal{D}(x,b)$$
i.e. can you prove
$$f(lambda x_1+(1-lambda)x_2)geq lambda f(x_1)+(1-lambda)f(x_2)$$
?
convex-analysis
asked Oct 26 '18 at 18:06
K. Sadri
696
Let $mathcal{S}$ denote a compact, convex set, $mathcal{B}$ be it's boundary and $mathcal{D}(.,.)$ a convex symmetric distance defined on it. i.e.
$$mathcal{D}(x,y)=mathcal{D}(y,x)$$
and
$$mathcal{D}(x, lambda y_1+(1-lambda)y_2)leqlambdamathcal{D}(x,y_1)+(1-lambda)mathcal{D}(x,y_2)$$
for all $lambdain[0,1]$ and $x,y,y_1,y_2inmathcal{S}$.
Is the following function concave?
$$f(x):=min_{binmathcal{B}}mathcal{D}(x,b)$$
i.e. can you prove
$$f(lambda x_1+(1-lambda)x_2)geq lambda f(x_1)+(1-lambda)f(x_2)$$
?
convex-analysis
convex-analysis
asked Oct 26 '18 at 18:06
K. Sadri
696
asked Oct 26 '18 at 18:06
K. Sadri
696
asked Oct 26 '18 at 18:06
K. Sadri
696
asked Oct 26 '18 at 18:06
K. Sadri
696
asked Oct 26 '18 at 18:06
K. Sadri
696
696
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
No, take $D(x,y) = |x-y|$ and $b=[0,1]$, then $f(x) = min{|x|, |x-1|}$ is neither convex nor concave.
answered Oct 26 '18 at 20:21
LinAlg
8,4161521
Nope, then $f(x)=min{x, 1-x}$ which is concave. however if you take $D(x,y)=(x-y)^2$ on the same set, then $f(x)=min{x^2, (1-x)^2}$ which is neither convex nor concave.
– K. Sadri
Nov 20 '18 at 10:08
@K.Sadri Sorry, I used $bin mathcal{S}$ instead of $bin mathcal{B}$. Your solution ($f(x)= min{x,1-x}$) allow for negative numbers. I fixed my initial example.
– LinAlg
Nov 20 '18 at 13:58
@K.Sadri consider accepting my answer to mark this question as answered
– LinAlg
Dec 7 '18 at 17:45
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%2f2972514%2fa-natural-concave-function%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
No, take $D(x,y) = |x-y|$ and $b=[0,1]$, then $f(x) = min{|x|, |x-1|}$ is neither convex nor concave.
answered Oct 26 '18 at 20:21
LinAlg
8,4161521
Nope, then $f(x)=min{x, 1-x}$ which is concave. however if you take $D(x,y)=(x-y)^2$ on the same set, then $f(x)=min{x^2, (1-x)^2}$ which is neither convex nor concave.
– K. Sadri
Nov 20 '18 at 10:08
@K.Sadri Sorry, I used $bin mathcal{S}$ instead of $bin mathcal{B}$. Your solution ($f(x)= min{x,1-x}$) allow for negative numbers. I fixed my initial example.
– LinAlg
Nov 20 '18 at 13:58
@K.Sadri consider accepting my answer to mark this question as answered
– LinAlg
Dec 7 '18 at 17:45
add a comment |
No, take $D(x,y) = |x-y|$ and $b=[0,1]$, then $f(x) = min{|x|, |x-1|}$ is neither convex nor concave.
answered Oct 26 '18 at 20:21
LinAlg
8,4161521
Nope, then $f(x)=min{x, 1-x}$ which is concave. however if you take $D(x,y)=(x-y)^2$ on the same set, then $f(x)=min{x^2, (1-x)^2}$ which is neither convex nor concave.
– K. Sadri
Nov 20 '18 at 10:08
@K.Sadri Sorry, I used $bin mathcal{S}$ instead of $bin mathcal{B}$. Your solution ($f(x)= min{x,1-x}$) allow for negative numbers. I fixed my initial example.
– LinAlg
Nov 20 '18 at 13:58
@K.Sadri consider accepting my answer to mark this question as answered
– LinAlg
Dec 7 '18 at 17:45
add a comment |
No, take $D(x,y) = |x-y|$ and $b=[0,1]$, then $f(x) = min{|x|, |x-1|}$ is neither convex nor concave.
answered Oct 26 '18 at 20:21
LinAlg
8,4161521
No, take $D(x,y) = |x-y|$ and $b=[0,1]$, then $f(x) = min{|x|, |x-1|}$ is neither convex nor concave.
answered Oct 26 '18 at 20:21
LinAlg
8,4161521
edited Nov 20 '18 at 13:57
answered Oct 26 '18 at 20:21
LinAlg
8,4161521
answered Oct 26 '18 at 20:21
LinAlg
8,4161521
answered Oct 26 '18 at 20:21
LinAlg
8,4161521
8,4161521
Nope, then $f(x)=min{x, 1-x}$ which is concave. however if you take $D(x,y)=(x-y)^2$ on the same set, then $f(x)=min{x^2, (1-x)^2}$ which is neither convex nor concave.
– K. Sadri
Nov 20 '18 at 10:08
@K.Sadri Sorry, I used $bin mathcal{S}$ instead of $bin mathcal{B}$. Your solution ($f(x)= min{x,1-x}$) allow for negative numbers. I fixed my initial example.
– LinAlg
Nov 20 '18 at 13:58
@K.Sadri consider accepting my answer to mark this question as answered
– LinAlg
Dec 7 '18 at 17:45
add a comment |
Nope, then $f(x)=min{x, 1-x}$ which is concave. however if you take $D(x,y)=(x-y)^2$ on the same set, then $f(x)=min{x^2, (1-x)^2}$ which is neither convex nor concave.
– K. Sadri
Nov 20 '18 at 10:08
@K.Sadri Sorry, I used $bin mathcal{S}$ instead of $bin mathcal{B}$. Your solution ($f(x)= min{x,1-x}$) allow for negative numbers. I fixed my initial example.
– LinAlg
Nov 20 '18 at 13:58
@K.Sadri consider accepting my answer to mark this question as answered
– LinAlg
Dec 7 '18 at 17:45
Nope, then $f(x)=min{x, 1-x}$ which is concave. however if you take $D(x,y)=(x-y)^2$ on the same set, then $f(x)=min{x^2, (1-x)^2}$ which is neither convex nor concave.
– K. Sadri
Nov 20 '18 at 10:08
Nope, then $f(x)=min{x, 1-x}$ which is concave. however if you take $D(x,y)=(x-y)^2$ on the same set, then $f(x)=min{x^2, (1-x)^2}$ which is neither convex nor concave.
– K. Sadri
Nov 20 '18 at 10:08
@K.Sadri Sorry, I used $bin mathcal{S}$ instead of $bin mathcal{B}$. Your solution ($f(x)= min{x,1-x}$) allow for negative numbers. I fixed my initial example.
– LinAlg
Nov 20 '18 at 13:58
@K.Sadri Sorry, I used $bin mathcal{S}$ instead of $bin mathcal{B}$. Your solution ($f(x)= min{x,1-x}$) allow for negative numbers. I fixed my initial example.
– LinAlg
Nov 20 '18 at 13:58
@K.Sadri consider accepting my answer to mark this question as answered
– LinAlg
Dec 7 '18 at 17:45
@K.Sadri consider accepting my answer to mark this question as answered
– LinAlg
Dec 7 '18 at 17:45
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%2f2972514%2fa-natural-concave-function%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
