Write in a concise form a set in 4-dimensional space. I have the following subset of the 4-dimensional space $(x_1,x_2,x_3,y)$:
\begin{align}
\mathcal I=&\left\{x_1=0\, , \, y\in[0, 1]\, , \, 0 \leq x_2\leq y\, , \, 0 \leq x_3\leq y\right\}\cup\\ &\left\{x_2=0\, , \, y\in[0, 1]\, , \, 0 \leq x_1\leq y\, , \, 0 \leq x_3\leq y\right\}\cup\\ &\left\{x_3=0\, , \, y\in[0, 1]\, , \, 0 \leq x_1\leq y\, , \, 0 \leq x_2\leq y\right\}\cup\\  &\left\{y=x_1\, , \, x_1\in[0,1] \, , \, 0 \leq x_2\leq x_1\, , \, 0 \leq x_3\leq x_1\right\}
\cup\\ &\left\{y=x_2\, , \, x_2\in[0,1] \, , \, 0\leq x_1\leq x_2\, , \, 0 \leq x_3\leq x_2\right\}
\cup\\ &\left\{y=x_3\, , \, x_3\in[0,1] \, , \, 0 \leq x_1\leq x_3\, , \, 0 \leq x_2\leq x_3\right\}
\end{align}
I would like to write $\mathcal I$ in a more concise form. Do you think it's possible? For example using a Cartesian product of sets.

How about
$$
{\cal{I}}=\{(x_1,x_2,x_3,y)\in[0,1]^4 : \max_i x_i \le y \;\wedge\; x_1x_2x_3(y-x_1)(y-x_2)(y-x_3)=0 \}
$$
?
