definition of simplex interior 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 simplex.
Target Context
- The reader will have a definition of simplex interior 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:
\( V\): \(\in \{\text{ the real vectors spaces }\}\)
\( \{p_0, ..., p_n\}\): \(\subseteq V\), \(\in \{\text{ the affine-independent sets of base points on } V\}\)
\( [p_0, ..., p_n]\): \(= \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\)
\( F\): \(= \{face_{S} ([p_0, ..., p_n]) \vert S \subset \{p_0, ..., p_n\}\}\), the set of the proper faces of \([p_0, ..., p_n]\)
\(*[p_0, ..., p_n]^\circ\): \(= [p_0, ..., p_n] \setminus \cup F\)
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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 \(V\), \(\{p_0, ..., p_n\}\), the affine simplex, \([p_0, ..., p_n] = \{\sum_{j = 0 \sim n} t^j p_j \in V \vert t^j \in \mathbb{R}, \sum_{j = 0 \sim n} t^j = 1 \land 0 \le t^j\}\), and the set of the proper faces of \([p_0, ..., p_n]\), \(F := \{face_{S} ([p_0, ..., p_n]) \vert S \subset \{p_0, ..., p_n\}\}\), \([p_0, ..., p_n]^\circ := [p_0, ..., p_n] \setminus \cup F\)
3: Note
The reason why it is qualified as "simplex interior" is that it can be different from 'topological interior': when a simplicial complex consists of a simplex and its proper faces, the topological interior of the simplex on the underlying space of the complex is the simplex itself, because the simplex is open on the underlying space, because the simplex is the whole underlying space and the whole underlying space is open on the whole underlying space; in the same case, the topological interior of any proper face of the simplex on the underlying space of the complex is empty, because no open subset of the underlying space of the complex is contained in the face.
