definition of ascending sequence of barycenters of faces 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
- 3: Note
Starting Context
- The reader knows a definition of ascending sequence of faces of affine simplex.
- The reader knows a definition of barycenter of affine simplex.
Target Context
- The reader will have a definition of ascending sequence of barycenters of faces 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:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\( [p_0, ..., p_n]\): \(= \text{ the affine simplex }\)
\( S\): \(= ([p_{\sigma_0}], [p_{\sigma_0}, p_{\sigma_1}], ..., [p_{\sigma_0}, ..., p_{\sigma_n}])\), \(\in \{\text{ the ascending sequences of faces of } [p_0, ..., p_n]\}\)
\(*B\): \(= (bary ([p_{\sigma_0}]), bary ([p_{\sigma_0}, p_{\sigma_1}]), ..., bary ([p_{\sigma_0}, ..., p_{\sigma_n}]))\)
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Conditions:
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2: Natural Language Description
For any real vectors space, \(V\), any affine-independent set of base points on \(V\), \(\{p_0, ..., p_n\} \subseteq V\), the affine simplex, \([p_0, ..., p_n]\), and any ascending sequence of faces of \([p_0, ..., p_n]\), \(S := ([p_{\sigma_0}], [p_{\sigma_0}, p_{\sigma_1}], ..., [p_{\sigma_0}, ..., p_{\sigma_n}])\), \(B := (bary ([p_{\sigma_0}]), bary ([p_{\sigma_0}, p_{\sigma_1}]), ..., bary ([p_{\sigma_0}, ..., p_{\sigma_n}]))\)
3: Note
"ascending sequence of barycenters" means that the set of the barycenters are given the order based on the order of the set of the faces whose barycenters its (of the set of the barycenters) elements are.
By those orders, the set of the barycenters of the simplexes in the simplicial complex, \(Vert Sd C\), is partially ordered: 1) irreflexive: \(\forall b \in Vert Sd C (\lnot b \lt b)\); 2) transitive: \(\forall b_1, b_2, b_3 \in Vert Sd C \text{ such that } b_1 \lt b_2 \land b_2 \lt b_3 (b_1 \lt b_3)\), because \(b_1 \lt b_2\) means that \(b_1\) and \(b_2\) are the barycenters of \(S_1\) and \(S_2\) where \(S_1\) is a face of \(S_2\) and \(b_2 \lt b_3\) means that \(b_2\) and \(b_3\) are the barycenters of \(S_2\) and \(S_3\) where \(S_2\) is a face of \(S_3\), which means that \(S_1\) is a face of \(S_3\), so, \(b_1 \lt b_3\). The reason why the set is denoted as "\(Vert Sd C\)" is that the simplicial complex (called barycentric subdivision of \(C\)), \(Sd C\), is created with the set as the vertexes set.
\(\{bary ([p_{\sigma_0}]), bary ([p_{\sigma_0}, p_{\sigma_1}]), ..., bary ([p_{\sigma_0}, ..., p_{\sigma_n}])\}\) is affine-independent, by the proposition that for any affine simplex and its any ascending sequence of faces, the set of the barycenters of the faces is affine-independent. So, each ascending sequence of barycenters of faces of each affine simplex or its each subsequence determines an affine simplex.
