Difference between revisions of "Simplex"

Latest revision as of 14:03, 31 January 2017

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

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

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

Notes

1. ↑ 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.
2. ↑ [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]
3. ↑ 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. ↑ ^{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