standard n-simplex
We know that a n-simplex is a convex hull of n+1 affinely independents points in $mathbb{R}^n$, i,e, let $ x_0,x_1, ldots ,x_n $ affinely independents points in $mathbb{R}^n$ then the n-simplex determinated is :
$$ C=lbrace lambda_0 x_0 + lambda_1 x_1+ ldots + lambda_n x_n / lambda_i geq 0, sum_{i=0}^{n}lambda_i =1 rbrace $$
Also a standard n-simplex, $Delta^n$, is the set :
$$ Delta^n=lbrace (t_0,t_1,ldots,t_n) in mathbb{R}^{n+1}/t_i geq 0, sum_{i=0}^{n}lambda_i =1 rbrace .$$
It is noted that the standard n-simplex is the convex hull of the canonical vectors $e_0,e_1, ldots, e_n in mathbb{R}^{n+1}$
We also see that the set $X=lbrace 0,e_0,e_1,ldots,e_n rbrace$ is affinely independent in $mathbb{R}^{n+1}$ so following the definition of n+1-simplex we should have that $Delta^n$ is a $n+1$-simplex because the convex hull of $X$ is $Delta^n$.
Why do the n-simplex standard define it that way if it is really a n + 1 simplex?
simplex
edited Dec 4 '18 at 1:37
Ethan Bolker
42k548111
asked Dec 4 '18 at 1:32
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
add a comment |
We know that a n-simplex is a convex hull of n+1 affinely independents points in $mathbb{R}^n$, i,e, let $ x_0,x_1, ldots ,x_n $ affinely independents points in $mathbb{R}^n$ then the n-simplex determinated is :
$$ C=lbrace lambda_0 x_0 + lambda_1 x_1+ ldots + lambda_n x_n / lambda_i geq 0, sum_{i=0}^{n}lambda_i =1 rbrace $$
Also a standard n-simplex, $Delta^n$, is the set :
$$ Delta^n=lbrace (t_0,t_1,ldots,t_n) in mathbb{R}^{n+1}/t_i geq 0, sum_{i=0}^{n}lambda_i =1 rbrace .$$
It is noted that the standard n-simplex is the convex hull of the canonical vectors $e_0,e_1, ldots, e_n in mathbb{R}^{n+1}$
We also see that the set $X=lbrace 0,e_0,e_1,ldots,e_n rbrace$ is affinely independent in $mathbb{R}^{n+1}$ so following the definition of n+1-simplex we should have that $Delta^n$ is a $n+1$-simplex because the convex hull of $X$ is $Delta^n$.
Why do the n-simplex standard define it that way if it is really a n + 1 simplex?
simplex
edited Dec 4 '18 at 1:37
Ethan Bolker
42k548111
asked Dec 4 '18 at 1:32
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
add a comment |
We know that a n-simplex is a convex hull of n+1 affinely independents points in $mathbb{R}^n$, i,e, let $ x_0,x_1, ldots ,x_n $ affinely independents points in $mathbb{R}^n$ then the n-simplex determinated is :
$$ C=lbrace lambda_0 x_0 + lambda_1 x_1+ ldots + lambda_n x_n / lambda_i geq 0, sum_{i=0}^{n}lambda_i =1 rbrace $$
Also a standard n-simplex, $Delta^n$, is the set :
$$ Delta^n=lbrace (t_0,t_1,ldots,t_n) in mathbb{R}^{n+1}/t_i geq 0, sum_{i=0}^{n}lambda_i =1 rbrace .$$
It is noted that the standard n-simplex is the convex hull of the canonical vectors $e_0,e_1, ldots, e_n in mathbb{R}^{n+1}$
We also see that the set $X=lbrace 0,e_0,e_1,ldots,e_n rbrace$ is affinely independent in $mathbb{R}^{n+1}$ so following the definition of n+1-simplex we should have that $Delta^n$ is a $n+1$-simplex because the convex hull of $X$ is $Delta^n$.
Why do the n-simplex standard define it that way if it is really a n + 1 simplex?
simplex
edited Dec 4 '18 at 1:37
Ethan Bolker
42k548111
asked Dec 4 '18 at 1:32
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
We know that a n-simplex is a convex hull of n+1 affinely independents points in $mathbb{R}^n$, i,e, let $ x_0,x_1, ldots ,x_n $ affinely independents points in $mathbb{R}^n$ then the n-simplex determinated is :
$$ C=lbrace lambda_0 x_0 + lambda_1 x_1+ ldots + lambda_n x_n / lambda_i geq 0, sum_{i=0}^{n}lambda_i =1 rbrace $$
Also a standard n-simplex, $Delta^n$, is the set :
$$ Delta^n=lbrace (t_0,t_1,ldots,t_n) in mathbb{R}^{n+1}/t_i geq 0, sum_{i=0}^{n}lambda_i =1 rbrace .$$
It is noted that the standard n-simplex is the convex hull of the canonical vectors $e_0,e_1, ldots, e_n in mathbb{R}^{n+1}$
We also see that the set $X=lbrace 0,e_0,e_1,ldots,e_n rbrace$ is affinely independent in $mathbb{R}^{n+1}$ so following the definition of n+1-simplex we should have that $Delta^n$ is a $n+1$-simplex because the convex hull of $X$ is $Delta^n$.
Why do the n-simplex standard define it that way if it is really a n + 1 simplex?
simplex
simplex
edited Dec 4 '18 at 1:37
Ethan Bolker
42k548111
asked Dec 4 '18 at 1:32
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
edited Dec 4 '18 at 1:37
Ethan Bolker
42k548111
asked Dec 4 '18 at 1:32
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
edited Dec 4 '18 at 1:37
Ethan Bolker
42k548111
edited Dec 4 '18 at 1:37
Ethan Bolker
42k548111
edited Dec 4 '18 at 1:37
Ethan Bolker
42k548111
42k548111
asked Dec 4 '18 at 1:32
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
asked Dec 4 '18 at 1:32
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
asked Dec 4 '18 at 1:32
Juan Daniel Valdivia FuentesJuan Daniel Valdivia Fuentes
104
104
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
The convex hull of $X$ is an ($n + 1$)-simplex, that's true, but it's not the same as $Delta^n$. For example, take $n = 1$. $Delta^n$ is the line segment from $(0, 1)$ to $(1, 0)$, but the convex hull of $X = {(0, 0), (1, 0), (0, 1)}$ is a solid triangle.
answered Dec 4 '18 at 3:23
Hew WolffHew Wolff
2,240716
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%2f3025004%2fstandard-n-simplex%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
The convex hull of $X$ is an ($n + 1$)-simplex, that's true, but it's not the same as $Delta^n$. For example, take $n = 1$. $Delta^n$ is the line segment from $(0, 1)$ to $(1, 0)$, but the convex hull of $X = {(0, 0), (1, 0), (0, 1)}$ is a solid triangle.
answered Dec 4 '18 at 3:23
Hew WolffHew Wolff
2,240716
add a comment |
The convex hull of $X$ is an ($n + 1$)-simplex, that's true, but it's not the same as $Delta^n$. For example, take $n = 1$. $Delta^n$ is the line segment from $(0, 1)$ to $(1, 0)$, but the convex hull of $X = {(0, 0), (1, 0), (0, 1)}$ is a solid triangle.
answered Dec 4 '18 at 3:23
Hew WolffHew Wolff
2,240716
add a comment |
The convex hull of $X$ is an ($n + 1$)-simplex, that's true, but it's not the same as $Delta^n$. For example, take $n = 1$. $Delta^n$ is the line segment from $(0, 1)$ to $(1, 0)$, but the convex hull of $X = {(0, 0), (1, 0), (0, 1)}$ is a solid triangle.
answered Dec 4 '18 at 3:23
Hew WolffHew Wolff
2,240716
The convex hull of $X$ is an ($n + 1$)-simplex, that's true, but it's not the same as $Delta^n$. For example, take $n = 1$. $Delta^n$ is the line segment from $(0, 1)$ to $(1, 0)$, but the convex hull of $X = {(0, 0), (1, 0), (0, 1)}$ is a solid triangle.
answered Dec 4 '18 at 3:23
Hew WolffHew Wolff
2,240716
edited Dec 4 '18 at 3:32
answered Dec 4 '18 at 3:23
Hew WolffHew Wolff
2,240716
answered Dec 4 '18 at 3:23
Hew WolffHew Wolff
2,240716
answered Dec 4 '18 at 3:23
Hew WolffHew Wolff
2,240716
2,240716
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%2f3025004%2fstandard-n-simplex%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
