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

2026-10-06

AI Summary

This study addresses the longstanding challenge in computational geometry of relatively approximating the mixed volume of a fixed number of convex bodies, where existing methods struggle to balance efficiency and accuracy. To overcome this limitation, this work proposes a randomized polynomial-time approximation algorithm based on membership queries. By integrating geometric probability analysis techniques, the method achieves efficient estimation of mixed volumes relying solely on a membership oracle. The primary contribution is the first breakthrough establishing complexity bounds that are polynomial in both the dimension and the precision parameter. With high probability, the algorithm outputs an ε-relative error estimate for any prescribed precision, thereby providing a novel paradigm for high-dimensional mixed volume computation that combines rigorous theoretical guarantees with practical feasibility.

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 $k$.

Problem

• mixed volumes
• convex bodies
• approximation algorithm
• membership oracles
• randomized algorithm

Innovation

• mixed volumes
• approximation algorithm
• randomized algorithm
• convex bodies
• membership oracle