Most of the algorithms for estimating the volume of a convex polyhedron $K \subset R^d$ assume the existence of an affine transform $T$ with the property that
$$ B \subset TK \tilde{\subset}\ \sigma B$$
where $B$ is the unit ball in $d$ dimensions, and $\sigma$ is $O(\sqrt{d})$.
(Update: the $\tilde{\subset}$ indicates that the containment is true except for an $\epsilon$-fraction of $K$)
The algorithms that I've seen for computing this transform are quite tricky. They require a bootstrap sampling process to extract a few points from inside $K$ which are then used to define the transformation.
However, the fact that such a transformation exists is folklore, and my question was:
Is there a simple algorithm (with possibly a weaker bound on $\sigma$) to compute the affine transform, given only a membership oracle for $K$ ?
