Convex functions: twice-differentiability property after partitioning domain into uncountable set
Suppose $f colon C rightarrow [0,1]$ is a convex function, where $C subseteq mathbb{R}^n$ is a full-dimensional compact convex set. A classical theorem of Alexandrov states that $f$ has second derivatives almost everywhere. That is, the set of points in int$(C)$ where $f$ is not twice differentiable forms a set of measure zero.
For all $epsilon in [0,1]$, let $C_{epsilon} := {x in C colon f(x) = epsilon}$. Certainly, $C = bigcup_{epsilon in [0,1]} C_{epsilon}$. Is the following statement true?
For almost every $epsilon in [0,1]$, $f$ has second derivatives almost everywhere in $C_{epsilon}$.
This question is probably best tackled using results from measure theory. However, each $C_{epsilon}$ could be of dimension less than $n$ (we partition $C$ into an uncountably infinite number of sets), so many standard results from measure theory do not apply.
To avoid some possibly trivial cases, it's fine if $C_{epsilon} neq emptyset$ for all $epsilon in [0,1]$.
measure-theory convex-analysis
asked Jan 3 at 20:58
ecto
1
New contributor
ecto is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
Suppose $f colon C rightarrow [0,1]$ is a convex function, where $C subseteq mathbb{R}^n$ is a full-dimensional compact convex set. A classical theorem of Alexandrov states that $f$ has second derivatives almost everywhere. That is, the set of points in int$(C)$ where $f$ is not twice differentiable forms a set of measure zero.
For all $epsilon in [0,1]$, let $C_{epsilon} := {x in C colon f(x) = epsilon}$. Certainly, $C = bigcup_{epsilon in [0,1]} C_{epsilon}$. Is the following statement true?
For almost every $epsilon in [0,1]$, $f$ has second derivatives almost everywhere in $C_{epsilon}$.
This question is probably best tackled using results from measure theory. However, each $C_{epsilon}$ could be of dimension less than $n$ (we partition $C$ into an uncountably infinite number of sets), so many standard results from measure theory do not apply.
To avoid some possibly trivial cases, it's fine if $C_{epsilon} neq emptyset$ for all $epsilon in [0,1]$.
measure-theory convex-analysis
asked Jan 3 at 20:58
ecto
1
New contributor
ecto is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
Suppose $f colon C rightarrow [0,1]$ is a convex function, where $C subseteq mathbb{R}^n$ is a full-dimensional compact convex set. A classical theorem of Alexandrov states that $f$ has second derivatives almost everywhere. That is, the set of points in int$(C)$ where $f$ is not twice differentiable forms a set of measure zero.
For all $epsilon in [0,1]$, let $C_{epsilon} := {x in C colon f(x) = epsilon}$. Certainly, $C = bigcup_{epsilon in [0,1]} C_{epsilon}$. Is the following statement true?
For almost every $epsilon in [0,1]$, $f$ has second derivatives almost everywhere in $C_{epsilon}$.
This question is probably best tackled using results from measure theory. However, each $C_{epsilon}$ could be of dimension less than $n$ (we partition $C$ into an uncountably infinite number of sets), so many standard results from measure theory do not apply.
To avoid some possibly trivial cases, it's fine if $C_{epsilon} neq emptyset$ for all $epsilon in [0,1]$.
measure-theory convex-analysis
asked Jan 3 at 20:58
ecto
1
New contributor
ecto is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Suppose $f colon C rightarrow [0,1]$ is a convex function, where $C subseteq mathbb{R}^n$ is a full-dimensional compact convex set. A classical theorem of Alexandrov states that $f$ has second derivatives almost everywhere. That is, the set of points in int$(C)$ where $f$ is not twice differentiable forms a set of measure zero.
For all $epsilon in [0,1]$, let $C_{epsilon} := {x in C colon f(x) = epsilon}$. Certainly, $C = bigcup_{epsilon in [0,1]} C_{epsilon}$. Is the following statement true?
For almost every $epsilon in [0,1]$, $f$ has second derivatives almost everywhere in $C_{epsilon}$.
This question is probably best tackled using results from measure theory. However, each $C_{epsilon}$ could be of dimension less than $n$ (we partition $C$ into an uncountably infinite number of sets), so many standard results from measure theory do not apply.
To avoid some possibly trivial cases, it's fine if $C_{epsilon} neq emptyset$ for all $epsilon in [0,1]$.
measure-theory convex-analysis
measure-theory convex-analysis
asked Jan 3 at 20:58
ecto
1
New contributor
ecto is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 3 at 20:58
ecto
1
New contributor
ecto is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 3 at 20:58
ecto
1
New contributor
ecto is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked Jan 3 at 20:58
ecto
1
asked Jan 3 at 20:58
ecto
1
1
New contributor
ecto is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
ecto is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
ecto is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
0
active
oldest
votes
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
});
}
});
ecto is a new contributor. Be nice, and check out our Code of Conduct.
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%2f3060992%2fconvex-functions-twice-differentiability-property-after-partitioning-domain-int%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
ecto is a new contributor. Be nice, and check out our Code of Conduct.
ecto is a new contributor. Be nice, and check out our Code of Conduct.
ecto is a new contributor. Be nice, and check out our Code of Conduct.
ecto is a new contributor. Be nice, and check out our Code of Conduct.
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%2f3060992%2fconvex-functions-twice-differentiability-property-after-partitioning-domain-int%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
