554: Face of Affine Simplex

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definition of face of affine simplex

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Natural Language Description

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Starting Context

• The reader knows a definition of affine simplex.

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Target Context

• The reader will have a definition of face of affine simplex.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$V$: $\in \{\text{ the real vectors spaces }\}$
$\{p_0,...,p_n\}$: $\subseteq V$, $\in \{\text{ the affine-independent sets of base points on }V\}$
$[p_0,...,p_n]$: $=\text{ the affine simplex }$
$\ast fac{e}_{\{j_1,...,j_l\}}([p_0,...,p_n])$: $=[p_0,...,\widehat{p_{j_1}},...,\widehat{p_{j_l}},...,p_n]$, where $\{j_1,...,j_l\}\subseteq \{0,...,n\}$ where $j_1<...<j_l$ and the hat mark denotes that the element is missing
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Conditions:
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$fac{e}_{\{j_1,...,j_l\}}([p_0,...,p_n])$ is called $(n-l)$-face of $[p_0,...,p_n]$ (there are $(n+1)!/(l!(n+1-l)!)$ $(n-l)$-faces).

When $0=l$, $fac{e}_{\{\}}([p_0,...,p_n])=[p_0,...,p_n]$ is a kind of face of $[p_0,...,p_n]$.

When $0<l$, $fac{e}_{\{j_1,...,j_l\}}([p_0,...,p_n])$ is called proper face of $[p_0,...,p_n]$.

$fac{e}_j([p_0,...,p_n]):=fac{e}_{\{j\}}([p_0,...,p_n])$ is called $j$-th face of $[p_0,...,p_n]$.

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2: Natural Language Description

For any real vectors space, $V$, and any affine-independent set of base points, $\{p_0,...,p_n\}\subseteq V$, and the affine simplex, $[p_0,...,p_n]$, any $(n-k)$-face of $[p_0,...,p_n]$, $fac{e}_{\{j_1,...,j_l\}}([p_0,...,p_n])$, is $[p_0,...,\widehat{p_{j_1}},...,\widehat{p_{j_l}},...,p_n]$, where $\{j_1,...,j_l\}\subseteq \{0,...,n\}$ where $j_1<...<j_l$ and the hat mark denotes that the element is missing

There are $(n+1)!/(l!(n+1-l)!)$ $(n-l)$-faces.

When $0=l$, $fac{e}_{\{\}}([p_0,...,p_n])=[p_0,...,p_n]$ is a kind of face of $[p_0,...,p_n]$.

When $0<l$, $fac{e}_{\{j_1,...,j_l\}}([p_0,...,p_n])$ is called proper face of $[p_0,...,p_n]$.

$fac{e}_j([p_0,...,p_n]):=fac{e}_{\{j\}}([p_0,...,p_n])$ is called $j$-th face of $[p_0,...,p_n]$.

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References

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