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The set of points $$\{(x_0,...,x_n)|\forall{i}: x_i \in [0,1], \ and \ x_0+..+x_n=1\}$$ is an n-simplex.

What can I call a set of points:

$$\{(x_0,...,x_n,y_0,...,y_n)|\forall{i}: x_i,y_i \in [0,1], \ and \ x_0+...+x_n = 1 \ and \ y_0+...+y_n=1\}$$ ?

for $n=1$, a 1-simplex is a line segment, and my shape is a square. This shape is like a "square simplex" - it is a simplex in $n$ dimensions, and a simplex in the other $n$ dimensions, but not in $2n$ dimensions.

Does this shape has a standard name? If not, what could be a good name for it?

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  • $\begingroup$ It looks like your set is an intersection of two faces of a simplex in $\mathbb{R}^{2n+2}$. $\endgroup$
    – Sigur
    Oct 30 '13 at 13:29
  • $\begingroup$ A product of simplices? $\endgroup$
    – Adam Saltz
    Oct 30 '13 at 18:12
  • $\begingroup$ @AdamSaltz Is this a cartesian product of simplices? $\endgroup$ Oct 30 '13 at 18:15
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    $\begingroup$ I'm not sure if that's a standard name for the construction. Using your definition (simplices are $n$-tuples satisfying some relation) then yes, this is the cartesian product of two simplices. $\endgroup$
    – Adam Saltz
    Oct 30 '13 at 18:18
  • $\begingroup$ @AdamSaltz thanks, "cartesian product of simplices" seems like a good term. $\endgroup$ Oct 30 '13 at 18:21
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What you call an"$n$-simplex" is in fact an $n$-dimensional hyperplane in $\Bbb R^{n+1}$. The second expression defines a $2n$-dimensional hyperplane in $\Bbb R^{2n+2}$.

Edit: Regarding your revised question: what you have isn't a type of simplex but a type of square (i.e. a square of a simplex). The term "simplex square" seems not to be laden with other mathematical meanings; so it would be one to consider.

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  • $\begingroup$ Sorry, I had a mistake in the definition of simplex. I corrected the question. I hope it is OK now. $\endgroup$ Oct 30 '13 at 17:49

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