let $D=\{(t_1,\cdots,t_d)\, |\, 0
an equivalent definition of a simplex
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$\begingroup$
general-topology
algebraic-topology
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0@Theo Buehler: I think there are two faces excluded the face $t_1=0$ and $t_d=1$ is my understanding correct? – 2011-07-17
1 Answers
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$\Delta^{d} = \{ (x_1, \dots, x_{d+1}) \in \mathbb R^{d+1} \;|\; \sum_{k=1}^{d+1} x_k = 1 \text{ and } 0 \leq x_1, \dots, x_{d+1} \leq 1 \}$ is probably the definition of a $d$-dimensional simplex you know.
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Let $t_i := \sum_{k=1}^{i} x_k$ for $1 \leq i \leq d+1$. We now have $0 \leq t_1 \leq \dots \leq t_d \leq t_{d+1} = 1$. Thus the faces defined by $x_1=0$ and $x_{d+1} = 0$ are excluded from $D$.
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0@Theo Buehler: Yes, that is what this site is for, and I interrupted your approach of doing so... @palio: Yes. Can you locate these faces in dimensions $1,2,3$ ? – 2011-07-17