Exposition of an approximation algorithm for mixed volumes of a constant number of convex bodies

Abstract

We study $\varepsilon$-relative approximation of mixed volumes of a fixed number $k$ of full-dimensional convex bodies in $\mathbb{R}^n$, given membership oracles and a known bound $B_n\subseteq K_i\subseteq R_0B_n$. We present a randomized algorithm that estimates any prescribed mixed volume within relative error $\varepsilon$ with probability at least $1-δ$, using polynomially many oracle calls and bit operations in $n$, $\log R_0$, $\varepsilon^{-1}$, and $\logδ^{-1}$ for fixed $k$.