It seems that it should be standard that the volume I am trying to compute should be known, but I was not successful in finding any reference. Anyway, I am trying to find the volume of the following:
Let $D,T > 0$ be parameters, and let $\textbf{u} = (u_1, \cdots, u_n), \textbf{v} =(v_1, \cdots, v_n) \in (\mathbb{R}^{+} \cup \{0\})^n$ be fixed vectors. Define the set $S(D,T,\textbf{u}, \textbf{v} )$ to be the intersection of the hyperplane section $\{(x_1, \cdots, x_n) \in \mathbb{R}^n : u_1 x_1 + \cdots + u_n x_n = D, x_i \geq 0, 1 \leq i \leq n\}$ and the set $\{(x_1, \cdots, x_n) \in \mathbb{R}^n : v_1 x_1 + \cdots + v_n x_n \leq T, x_i \geq 0, 1 \leq i \leq n\}$. Let $A(S) = A(S(D,T,\textbf{u}, \textbf{v}))$ denote the volume of $S$. My questions are:
1) Is there an explicit formula to compute $A(S)$? which depends only on $D,T, \textbf{u}, \textbf{v}$?
2) If such a general formula is not available, can one compute the value of $A(S)$ which is maximal for fixed $T, \textbf{u}, \textbf{v}$ and allowing $D$ to vary?
Thanks for any insights.