We are given rational numbers $[c_1, c_2, \ldots, c_n]$ and $v$ from the interval $[0,1]$.
Consider the $n$-fold integral $$ J = \int_{\theta_1 \in I_1, \theta_2 \in I_2 \ldots, \theta_n \in I_n} d\theta_n\ldots d\theta_2 d\theta_1 $$ whose intervals are defined by $$ I_j = \begin{cases} [0,1] & j = 1\\ [\max(c_j,\theta_{j-1}),1] & 2\leq j\leq n \end{cases} $$ We want to check if $J$ is equal to $v$.
Is this problem $\mathsf{NP}$-hard?
Informally, each $\max$ in the lower limits of the intervals leads to a two-way split in evaluating the integral, and thus to $2^{n-1}$ integrals that sum to $J$.
Note that $J$ is also the volume of an $n$ dimensional polytope define by the following inequalities:
$$c_j \leq x_j \leq 1 \ (\text{for } 1 \leq j \leq n),$$ $$0 \leq x_1 \leq x_2 \leq \ldots \leq x_n \leq 1.$$
Is there a result in computational complexity or computational geometry about volume of such shapes?
