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]^{[Note 1]}. We define the "[ilmath]n[/ilmath]-simplex", [ilmath]\sigma[/ilmath], spanned by [ilmath]\{a_0,\ldots,a_n\} [/ilmath] to be the following set^{[1]}:
- [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
- [ilmath]\sigma[/ilmath] is convex
- 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^{[1]} 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]^{[1]}
- Vertices: the vertices of [ilmath]\sigma[/ilmath] are the points [ilmath]a_0,\ldots,a_n[/ilmath]^{[1]}
- Face: any simplex spanned by [ilmath]A\in\big(\mathcal{P}(\{a_0,\ldots,a_n\})-\{\emptyset\}\big)[/ilmath] is called a face^{[1]} of [ilmath]\sigma[/ilmath].
- The face is a "proper face"^{[1]} if it is not [ilmath]\sigma[/ilmath] itself^{[Note 2]}.
- 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]^{[Note 3]}
- Boundary: the union of all proper faces is the boundary of [ilmath]\sigma[/ilmath]^{[1]} denoted [ilmath]\text{Bd}(\sigma)[/ilmath]^{[1]} 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]^{[1]} - 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
Notes
- ↑ This means that [ilmath]N > n[/ilmath] - certainly. We may be able to go lower (to [ilmath]N\ge n[/ilmath]) but I don't want to at this time.
- ↑ [ilmath]\sigma[/ilmath] is a face of itself. Much like [ilmath]\{a,b\}\subseteq\{a,b\} [/ilmath] is a subset, but [ilmath]\{a\}\subset\{a,b\} [/ilmath] is a proper subset of [ilmath]\{a,b\} [/ilmath]
- ↑ It is quite common to denote "deletion from the array" by a [ilmath]\hat{a} [/ilmath] for [ilmath]a\in[/ilmath] the array
References
- ↑ ^{1.0} ^{1.1} ^{1.2} ^{1.3} ^{1.4} ^{1.5} ^{1.6} ^{1.7} ^{1.8} Elements of Algebraic Topology - James R. Munkres