Find the boundary of $\mathcal{D}_n:=\left\{(x_1,\dots,x_n,y)\, \big|\, 0\leq x_i\leq y\, ,\forall i=1,\dots,n,\, y\in[0,1]\right\}\, .$

Question

Fix $n\in\mathbb N$. What is the boundary $\partial D_n$ of the following set? $$\mathcal{D}_n:=\left\{(x_1,\dots,x_n,y)\, \big|\, 0\leq x_i\leq y\, ,\forall i=1,\dots,n,\, y\in[0,1]\right\}\, .$$ When $n=1$, we have $$\partial D_1=\left\{x_1=0\, , \, y\in[0,1]\right\}\cup\left\{y=1\, , \, x\in[0,1]\right\}\cup \left\{y=x_1\, , \, x\in[0,1]\right\}$$ For $n>1$?

Answer 1

You may want to draw this out. In $\mathbb{R}^2$, consider the unit square I$_2$. D$_1$ consists of the lower diagonal of the square. In other words, it's the intersection of {(x,y) $\mid$ x $\le$ y} and I$_2$. I think if you generalize this, you get a similar object.

So the boundary is the plane $x_1=x_2\ldots=x_{n+1}$ together with the boundary faces of the unit square in $\mathbb{R}^{n+1}$ that lie in the "lower" half-space.