An $\mathsf{FPRAS}$ for Mixed Volumes

Abstract

We design the first fully polynomial-time randomized approximation scheme ($\mathsf{FPRAS}$) to estimate the mixed volume of $n$ arbitrary compact convex sets in $\mathbb{R}^{n}$ specified by weak membership oracles. This disproves a longstanding conjecture of Gurvits (2009). $\textbf{AI Disclosure}$ The main ideas were generated entirely by GPT 6 Astra. The purpose of this paper is to provide a human-written exposition of the algorithm.