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Exact Volumes of Semi-Algebraic Convex Bodies

Lakshmi Ramesh, Nicolas Weiss

TL;DR

The authors address exact volume computation for semi-algebraic convex bodies defined by concave polynomials by representing volumes as periods and solving the resulting Picard–Fuchs differential equations within a holonomic $D$-module framework. Their key innovation is a method to identify the two relevant critical values for any projection, which reduces the number of creative telescoping steps exponentially and converts the recursion into a path-like computation. They provide a complete SageMath implementation leveraging ore_algebra, Macaulay2, msolve, and HypersurfaceRegions.jl, and demonstrate high-precision volumes for problems such as intersections of translated $\ell_p$-balls, thereby enabling precise geometric statistics computations. The work advances deterministic, high-precision volume computation for convex semi-algebraic sets and highlights practical considerations in choosing projections and handling critical-value computations.

Abstract

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. This is applied to a problem in geometric statistics, where the convex body is an intersection of $\ell_p$-balls.

Exact Volumes of Semi-Algebraic Convex Bodies

TL;DR

The authors address exact volume computation for semi-algebraic convex bodies defined by concave polynomials by representing volumes as periods and solving the resulting Picard–Fuchs differential equations within a holonomic -module framework. Their key innovation is a method to identify the two relevant critical values for any projection, which reduces the number of creative telescoping steps exponentially and converts the recursion into a path-like computation. They provide a complete SageMath implementation leveraging ore_algebra, Macaulay2, msolve, and HypersurfaceRegions.jl, and demonstrate high-precision volumes for problems such as intersections of translated -balls, thereby enabling precise geometric statistics computations. The work advances deterministic, high-precision volume computation for convex semi-algebraic sets and highlights practical considerations in choosing projections and handling critical-value computations.

Abstract

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. This is applied to a problem in geometric statistics, where the convex body is an intersection of -balls.
Paper Structure (10 sections, 7 theorems, 47 equations, 3 figures, 3 algorithms)

This paper contains 10 sections, 7 theorems, 47 equations, 3 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Volumes via differential equations
  3. Notation.
  4. Convex Sets and Their Deformations
  5. CriticalValues
  6. Case 1 $(\dim({\operatorname{CritPts}}) = 0)$:
  7. Case 2 $(\operatorname{dim}({\operatorname{CritPts}})>0)$:
  8. 1DimVolume
  9. SuitableValues
  10. Implementation and experiments

Key Result

lemma 1

Let $C = \{ x \in \mathbb{R}^n \mid f(x) > 0 \}$ be a bounded region of $\mathbb R^n$ given by a single polynomial $f\in \mathbb{Q}[x_1,\ldots, x_n]$ and assume that its vanishing set $V(f)$ is a smooth variety. Then, the volume of the semi-algebraic set $C$ is a period of the rational function That is, where $\Gamma$ is a closed cycle $\Gamma \subset \mathbb{C}^n-\partial C$.

Figures (3)

  • Figure 1: Two $\ell_4$-balls deformed by a parameter $t$
  • Figure 2: Convex set with non-convex deformation
  • Figure 3: Sampling points to select the correct critical values

Theorems & Definitions (11)

  • lemma 1
  • definition 1
  • theorem 1
  • proof : Proof of Theorem \ref{['thm:takayama']}
  • proposition 1: Lairez_volumes_semi_algebraic_sets
  • lemma 2
  • proposition 2: Lairez_volumes_semi_algebraic_sets
  • lemma 3
  • proof
  • proposition 3
  • ...and 1 more