I have a simple question. Is it possible to construct a 3-simplex (a, b, c, d) in the following figure?
My guess was that as we can generate an edge (a, c) by a linear combination of (a, d) and (a, b), there is no 3-simplex in this figure. It's rather two 2-simplices?
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Def: n-simplex
Let $\{p_0, \dots, p_n \}$ be a geometrically independent set in $\mathbb{R}^d$. The n-simplex $\sigma$ spanned by the points $p_i$ to be the set of all points $x \in \mathbb{R}^d$ of the form; $x = \sum^n_{i=0} t_i p_i$ where $\sum^n_{i=0} t_i = 1$ with $t_i$ being non-negative.