definition 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 affine-independent set of points on real vectors space.
Target Context
- The reader will have a definition 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:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\(*[p_0, ..., p_n]\): \(= \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\)
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Conditions:
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\([p_0, p_1, ..., p_n]\) is called affine n-simplex.
2: Natural Language Description
For any real vectors space, \(V\), and any affine-independent set of base points, \(\{p_0, ..., p_n\} \subseteq V\), the set, \([p_0, ..., p_n] := \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\)
\([p_0, ..., p_n]\) is called affine n-simplex.
3: Note
While it is usually defined with \(V = \mathbb{R}^d\), the Euclidean vectors space with the Euclidean topology, this definition made the more general definition with any general real vectors space. There is no critical difference between them, because with any \(d\)-dimensional \(V\) equipped with the canonical topology, \(V\) is homeomorphic to \(\mathbb{R}^d\), and the subspace, \([p_0, ..., p_n] \subseteq V\), is homeomorphic to the subspace, \([p_0, ..., p_n] \subseteq \mathbb{R}^d\).
