Request for verification of the technique for deducing vertices of a polytope

Question

I am trying to find the vertices of a polytope which is defined by two parallel planes of the form ${\bf{a}^T\bf{w}}=c_1, {\bf{a}^T\bf{w}}=c_2$, where $\bf{a}\in \mathbb{R}^n_+, \bf{w}\in \mathbb{R}^n_+$, and $c_1,c_2>0$, as well as the constraints $w_1\ge w_2\ge \cdots w_n\ge 0$. I found the vertices, as $(u_1,0,\cdots,0),(u_2,u_2,0,\cdots,0),\cdots,(u_n,u_n,\cdots,u_n)$, and $(v_1,0,\cdots,0),(v_2,v_2,0,\cdots,0),\cdots,(v_n,v_n,\cdots,v_n)$, where $v_k=c_2/c_1u_k$, and the $u_1,\cdots,u_n$ are obtained by taking intersections of $w_1\ge w_2\ge \cdots \ge w_n\ge 0$ with ${\bf{w}^T\bf{a}}=c_1$. Can anyone kindly verify whether my approach is correct? Thanks in advance.

Answer 1

Yes, the cone $w_1\ge w_2 \ge \dots\ge w_n \ge 0$ is spanned by the ray generators $(1,0,0,\dots,0)$, $(1,1,0,\dots,0)$, …, $(1,1,1,\dots,1)$. Intersecting these rays with your two parallel hyperplanes yields the vertices of the polytope.