We study in this paper randomized algorithms to approximate the mixed volume of well-presented convex compact sets.
Our main result is a poly-time algorithm which approximates $V(K_1,...,K_n)$ with multiplicative error $e^n$ and
with better rates if the affine dimensions of most of the sets $K_i$ are small.\\
Our approach is ... more >>>

Let $X$ be randomly chosen from $\{-1,1\}^n$, and let $Y$ be randomly
chosen from the standard spherical Gaussian on $\R^n$. For any (possibly unbounded) polytope $P$
formed by the intersection of $k$ halfspaces, we prove that
$$\left|\Pr\left[X \in P\right] - \Pr\left[Y \in P\right]\right| \leq \log^{8/5}k ... more >>>