Simplex

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I don't like the ambient [ilmath]\mathbb{R}^N[/ilmath] requirement, look to drop it. Also get more references

Definition

Let [ilmath]\{a_0,\ldots,a_n\}\subseteq\mathbb{R}^N[/ilmath] be a geometrically independent set in [ilmath]\mathbb{R}^N[/ilmath]. We define the "[ilmath]n[/ilmath]-simplex", [ilmath]\sigma[/ilmath], spanned by [ilmath]\{a_0,\ldots,a_n\} [/ilmath] to be the following set:

• [math]\sigma:\eq\left\{\ x\in\mathbb{R}^N\ \left\vert\ x\eq\sum^n_{i\eq 0}t_ia_i\wedge (t_i)_{i\eq 0}^n\subset\mathbb{R}\wedge \forall i\in\{0,\ldots,n\}\subset\mathbb{N}[t_i\ge 0]\wedge\sum_{i\eq 0}^n t_i\eq 1\right\}\right. [/math]

The numbers, [ilmath](t_i)_{i\eq 0}^n\subset\mathbb{R} [/ilmath] are uniquely determined by [ilmath]x[/ilmath] and are called the "barycentric coordinates" of [ilmath]x[/ilmath] with respect to [ilmath]\{a_0,\ldots,a_n\} [/ilmath]

Elementary properties

1. [ilmath]\sigma[/ilmath] is convex
2. In fact [ilmath]\sigma[/ilmath] is the convex hull of [ilmath]\{a_0,\ldots,a_n\} [/ilmath]

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See page 6 of there's a list of 6 points. Could be useful

Terminology

• Dimension: [ilmath]\text{Dim}(\sigma):\eq\vert\{a_0,\ldots,a_n\}\vert-1[/ilmath]
• Vertices: the vertices of [ilmath]\sigma[/ilmath] are the points [ilmath]a_0,\ldots,a_n[/ilmath]
• Face: any simplex spanned by [ilmath]A\in\big(\mathcal{P}(\{a_0,\ldots,a_n\})-\{\emptyset\}\big)[/ilmath] is called a face of [ilmath]\sigma[/ilmath].
  • The face is a "proper face" if it is not [ilmath]\sigma[/ilmath] itself.
  • TODO: I do not think the empty set is a face. I think Munkres was just being lax, check this
• Face opposite [ilmath]a_i\in\{a_0,\ldots,a_n\} [/ilmath]: is the face spanned by [ilmath]\{a_0,\ldots,a_n\}-\{a_i\} [/ilmath] which is sometimes denoted [ilmath]\{a_0,\ldots,\hat{a_i},\ldots,a_n\} [/ilmath]
• Boundary: the union of all proper faces is the boundary of [ilmath]\sigma[/ilmath] denoted [ilmath]\text{Bd}(\sigma)[/ilmath] or [ilmath]\partial\sigma[/ilmath]
  • i.e. [ilmath]\partial\sigma:\eq \bigcup_{\tau\in K }\tau [/ilmath] where [ilmath]K:\eq \big(\mathcal{P}(\{a_0,\ldots,a_n\})-\{\emptyset,\{a_0,\ldots,a_n\}\}\big) [/ilmath]
• Interior: [ilmath]\text{Int}(\sigma):\eq\sigma-\partial\sigma[/ilmath] - the interior is sometimes called an open simplex

Proof of claims

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The claim that the coordinates are unique and stuff is totally missing