  Given an $n$-dimensional convex body by a membership oracle in general, it is
known that any polynomial-time deterministic algorithm cannot approximate its
volume within ratio $(n/\log n)^n$. There is a substantial progress on
randomized approximation such as Markov chain Monte Carlo for a
high-dimensional volume, and for many #P-hard problems, while some
deterministic approximation algorithms are recently developed only for a few
#P-hard problems. Motivated by a deterministic approximation of the volume of a
${\cal V}$-polytope, that is a polytope with few vertices and (possibly)
exponentially many facets, this paper investigates the volume of a "knapsack
dual polytope," which is known to be #P-hard due to Khachiyan (1989). We reduce
an approximate volume of a knapsack dual polytope to that of the intersection
of two cross-polytopes, and give FPTASs for those volume computations.
Interestingly, the volume of the intersection of two cross-polytopes (i.e.,
$L_1$-balls) is #P-hard, unlike the cases of $L_{\infty}$-balls or $L_2$-balls.
