I am given a set of inequalities $v_1\ge v_2\ge \cdots\ge v_n\ge 0$, and a set of bounds for the coordinates: $v_k\in[0,a_k],\ 1\le k\le n$, where $a_1\ge a_2\ge \cdots\ge a_n\ge 0$. My question is:
What are the vertices of the polytope created by these set of constraints?
I think that the coordinates of the vertices are going to be given by $(0,0,\cdots,0),(a_1,0,\cdots,0),(a_2,a_2,0,\cdots,0),\cdots,(a_n,a_n,\cdots,a_n)$, and $(a_1,a_2,\cdots,a_n)$. I could not rigorously find this, but guessed this should be the result after finding the coordinates for 2 dimension, and interpolating the result for higher dimensions. However, I want to know, first of all, whether this result is correct, and how to find this result rigorously for any dimension. Please help me out.
This is a marked order polytope as introduced by Ardila, Bliem and Salazar as a generalization of the order polytope by Stanley. The face structure (in particular vertices) is discussed in this article in terms of partitions of the underlying poset.
In your case I think you are missing vertices: for $n=4$ you get a vertex $(a_2,a_2,a_4,a_4)$ for example.
Letting $a_{n+1}=0$, the full list of vertices should be those points of the form $(a_{i_1},a_{i_2},\dots,a_{i_n})$ with indices $i_k\in\{1,\dots,n+1\}$ satisfying
A constructive description of these is the following: In the last coordinate pick either $0$ or $a_n$, then inductively for the $k$-th coordinate with $k=n-1,n-2,...,1$ pick either the value you picked for $k+1$ or pick $a_k$.
By this description we have $2^n$ vertices, assuming that the $a_j$ are all distinct.
Note that this example is also unimodularly equivalent to the Stanley–Pitman polytope, as discussed in this MO thread.