Exact Volumes of Semi-Algebraic Convex Bodies
This work provides a novel computational tool for exact volume computation of convex semi-algebraic sets, benefiting researchers in algebraic geometry and convex optimization.
The paper presents a method to compute volumes of semi-algebraic convex bodies defined by concave polynomial inequalities to arbitrary precision, leveraging linear differential equations and reducing creative telescoping steps exponentially due to convexity. Examples in 2, 3, and 4 dimensions are provided.
We compute the volumes of convex bodies that are given by inequalities of concave polynomials. These volumes are found to arbitrary precision thanks to the representation of periods by linear differential equations. Our approach rests on work of Lairez, Mezzarobba, and Safey El Din. We present a novel method to identify the relevant critical values. Convexity allows us to reduce the required number of creative telescoping steps by an exponential factor. We provide an implementation based on the ore_algebra package in SageMath. We present examples computed with our implementation in 2, 3 and 4 dimensions.