definition of face of affine simplex
Topics
About: vectors space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description
- 2: Natural Language Description
Starting Context
- The reader knows a definition of affine simplex.
Target Context
- The reader will have a definition of face of affine simplex.
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.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\( [p_0, ..., p_n]\): \(= \text{ the affine simplex }\)
\(*face_{\{j_1, ..., j_l\}} ([p_0, ..., p_n])\): \(= [p_0, ..., \hat{p_{j_1}}, ..., \hat{p_{j_l}}, ..., p_n]\), where \(\{j_1, ..., j_l\} \subseteq \{0, ..., n\}\) where \(j_1 \lt ... \lt j_l\) and the hat mark denotes that the element is missing
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Conditions:
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\(face_{\{j_1, ..., j_l\}} ([p_0, ..., p_n])\) is called \((n - l)\)-face of \([p_0, ..., p_n]\) (there are \((n + 1)! / (l! (n + 1 - l)!)\) \((n - l)\)-faces).
When \(0 = l\), \(face_{\{\}} ([p_0, ..., p_n]) = [p_0, ..., p_n]\) is a kind of face of \([p_0, ..., p_n]\).
When \(0 \lt l\), \(face_{\{j_1, ..., j_l\}} ([p_0, ..., p_n])\) is called proper face of \([p_0, ..., p_n]\).
\(face_j ([p_0, ..., p_n]) := face_{\{j\}} ([p_0, ..., p_n])\) is called \(j\)-th face of \([p_0, ..., p_n]\).
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 \([p_0, ..., p_n]\), \(face_{\{j_1, ..., j_l\}} ([p_0, ..., p_n])\), is \([p_0, ..., \hat{p_{j_1}}, ..., \hat{p_{j_l}}, ..., p_n]\), where \(\{j_1, ..., j_l\} \subseteq \{0, ..., n\}\) where \(j_1 \lt ... \lt j_l\) and the hat mark denotes that the element is missing
There are \((n + 1)! / (l! (n + 1 - l)!)\) \((n - l)\)-faces.
When \(0 = l\), \(face_{\{\}} ([p_0, ..., p_n]) = [p_0, ..., p_n]\) is a kind of face of \([p_0, ..., p_n]\).
When \(0 \lt l\), \(face_{\{j_1, ..., j_l\}} ([p_0, ..., p_n])\) is called proper face of \([p_0, ..., p_n]\).
\(face_j ([p_0, ..., p_n]) := face_{\{j\}} ([p_0, ..., p_n])\) is called \(j\)-th face of \([p_0, ..., p_n]\).
