description/proof of that for affine simplex, ascending sequence of faces, and set of barycenters of faces, convex combination of subset of set of barycenters is convex combination w.r.t. set of vertexes 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: Proof
- 4: Note
Starting Context
- The reader knows a definition of affine simplex.
- The reader knows a definition of ascending sequence of faces of affine simplex.
- The reader knows a definition of barycenter of affine simplex.
- The reader knows a definition of convex combination of possibly-non-affine-independent set of base points on real vectors space.
Target Context
- The reader will have a description and a proof of the proposition that for any affine simplex, its any ascending sequence of faces, and the set of the barycenters of the faces, any convex combination of any subset of the set of the barycenters is a convex combination with respect to the set of the vertexes of the 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:
\(V\): \(\in \{\text{ the real vectors spaces }\}\)
\(\{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\([p_0, ..., p_n]\): \(= \text{ the affine simplex }\)
\(S\): \(= ([p'_0], [p'_0, p'_1], ..., [p'_0, ..., p'_n])\), \(\in \{\text{ the ascending sequences of faces of } [p_0, ..., p_n]\}\)
\(S'\): \(= \{bary ([p'_0]), bary ([p'_0, p'_1]), ..., bary ([p'_0, ..., p'_n])\} = \{b_0, ..., b_n\}\)
\(S''\): \(= \{b_{k_0}, ..., b_{k_l}\}\), \(\subseteq S'\)
\(p\): \(= t^0 b_{k_0} + ... + t^k b_{k_l}\), where \(\sum_{j \in \{0, ..., l\}} t^j = 1\) and \(0 \le t^j\)
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Statements:
\(p = t'^0 p_0 + ... + t'^n p_n\), where \(\sum_{j \in \{0, ..., n\}} t'^j = 1\) and \(0 \le t'^j\).
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2: Natural Language Description
For any real vectors space, \(V\), any affine-independent set of base points on \(V\), \(\{p_0, ..., p_n\}\), the affine simplex, \([p_0, ..., p_n]\), any ascending sequence of faces of \([p_0, ..., p_n]\), \(S = ([p'_0], [p'_0, p'_1], ..., [p'_0, ..., p'_n])\), the set of the barycenters of the faces, \(S' = \{bary ([p'_0]), bary ([p'_0, p'_1]), ..., bary ([p'_0, ..., p'_n])\} = \{b_0, ..., b_n\}\), and any subset, \(S'' = \{b_{k_0}, ..., b_{k_l}\} \subseteq S'\), of \(S'\), any convex combination of \(S''\), \(p = t^0 b_{k_0} + ... + t^k b_{k_l}\), where \(\sum_{j \in \{0, ..., l\}} t^j = 1\) and \(0 \le t^j\), is a convex combination of the vertexes of \([p_0, ..., p_n]\), which is \(p = t'^0 p_0 + ... + t'^n p_n\), where \(\sum_{j \in \{0, ..., n\}} t'^j = 1\) and \(0 \le t'^j\).
3: Proof
\(b_{k_j} = 1 / (k_j + 1) (p'_0 + ... + p'_{k_j})\).
\(p = t^0 1 / (k_0 + 1) (p'_0 + ... + p'_{k_0}) + ... + t^l 1 / (k_l + 1) (p'_0 + ... + p'_{k_l}) = (t^0 1 / (k_0 + 1) + ... + t^l 1 / (k_l + 1)) p'_0 + ... + t^l 1 / (k_l + 1) p'_{k_l}\).
Each coefficient of \(p'_{k_j}\) is non-negative.
The sum of the coefficients is \((t^0 1 / (k_0 + 1) + ... + t^l 1 / (k_l + 1)) + ... + t^l 1 / (k_l + 1) = (t^0 1 / (k_0 + 1) + ... + t^l 1 / (k_l + 1)) (k_0 + 1) + (t^1 1 / (k_1 + 1) + ... + t^l 1 / (k_l + 1)) (k_1 + 1 - (k_0 + 1)) + ... + t^l 1 / (k_l + 1) (k_l + 1 - (k_{l - 1} + 1))\), where the multiplication by \((k_0 + 1)\) is for \(p'_0, ..., p'_{k_0}\), the multiplication by \(k_1 + 1 - (k_0 + 1)\) is for \(p'_{k_0 + 1}, ..., p'_{k_1}\), etc.. \(= 1 / (k_0 + 1) (k_0 + 1) t^0 + (1 / (k_1 + 1) (k_0 + 1) + 1 / (k_1 + 1) (k_1 + 1 - (k_0 + 1))) t^1 + ... + (1 / (k_l + 1) (k_0 + 1) + 1 / (k_l + 1) (k_1 + 1 - (k_0 + 1)) + ... + 1 / (k_l + 1) (k_l + 1 - (k_{l - 1} + 1))) t^l = t^0 + ... + t^l = 1\).
Note that \((p'_0, ..., p'_n)\) is just a sequence of \(\{p_0, ..., p_n\}\).
4: Note
This proposition, 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, and the proposition that any subset of any affine-independent set of points on any real vectors space is affine-independent implies that \([b_{k_0}, ..., b_{k_l}] \subseteq [p_0, ..., p_n]\).
