# term for a "squared simplex"

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?

• It looks like your set is an intersection of two faces of a simplex in $\mathbb{R}^{2n+2}$. Oct 30 '13 at 13:29
• A product of simplices? Oct 30 '13 at 18:12
• @AdamSaltz Is this a cartesian product of simplices? Oct 30 '13 at 18:15
• 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. Oct 30 '13 at 18:18
• @AdamSaltz thanks, "cartesian product of simplices" seems like a good term. Oct 30 '13 at 18:21

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 $\Bbb R^{2n+2}$.