I have a question regarding the definition of a simplex. I am quoting the definition of a simplex from Wikipedia, which is similar to the definition in my textbook too

"Specifically, a $k$-simplex is a $k$-dimensional polytope which is the convex hull of its $k + 1$ vertices. More formally, suppose the $k + 1$ points $u_0,\dots, u_k \in \mathbb{R}^n$ are affinely independent, which means $u_1 - u_0,\dots, u_k-u_0$ are linearly independent. Then, the simplex determined by them is the set of points $C =\{\theta_0 u_0 + \dots+\theta_k u_k | \theta_i \ge 0, 0 \le i \le k, \sum_{i=0}^{k} \theta_i=1\} $.

Can we define a simplex using $k$ points that are linearly independent instead of $k+1$ points that are affinely independent? Is it the same thing or am I missing something?

Thanks in advance!


You could replace each $u_i$ with $u_i-u_0$, so you would have $k$ linearly independent points plus the origin.

This is likely to be less convenient though. In any simplicial complex that isn't just a 1-simplex there is no point that is shared by all the simplices, so the different simplices would have to be defined with respect to different origins. (Edit: It's possible that this isn't an issue in certain contexts. I'm coming at this from a topology perspective, and have essentially no knowledge of convex analysis.)


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