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How to compute the volume in low dimension?

Arjan Cornelissen, Simon Apers, Sander Gribling

TL;DR

This work analyzes volume estimation for convex bodies in the low-dimensional, high-precision regime via a membership oracle, deriving tight query-complexity characterizations across deterministic, randomized, and quantum models. It shows that convex set estimation and $\varepsilon$-kernel construction admit $\widetilde{\Theta}(\varepsilon^{-(d-1)/2})$ membership-queries irrespective of randomness or quantum power, while volume estimation benefits from randomness and quantum speedups: $\widetilde{O}(\varepsilon^{-2(d-1)/(d+3)})$ (randomized) and $\widetilde{O}(\varepsilon^{-(d-1)/(d+1)})$ (quantum). The key techniques combine a deterministic rounding to obtain a well-rounded body, projection-based $\varepsilon$-kernel construction, and inner/outer approximations with sampling or amplitude estimation for volume, together with Dudley-type spherical-cap packings to prove matching lower bounds. The results provide a complete low-dimensional theory for volume estimation with membership oracles and clarify the potential of quantum and randomized accelerations in this setting, complementing the extensive high-dimensional literature. Overall, the work reveals sharp, dimension-dependent trade-offs: randomness and quantum advantages vanish for convex-set tasks but yield meaningful reductions for volume estimation in fixed dimension.

Abstract

Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the query complexity of volume estimation to a membership oracle has been a longstanding open question. Most of the previous work focuses on the high-dimensional limit. In this work, we tightly characterize the deterministic, randomized and quantum query complexity of this problem in the high-precision limit, i.e., when the dimension is constant.

How to compute the volume in low dimension?

TL;DR

This work analyzes volume estimation for convex bodies in the low-dimensional, high-precision regime via a membership oracle, deriving tight query-complexity characterizations across deterministic, randomized, and quantum models. It shows that convex set estimation and -kernel construction admit membership-queries irrespective of randomness or quantum power, while volume estimation benefits from randomness and quantum speedups: (randomized) and (quantum). The key techniques combine a deterministic rounding to obtain a well-rounded body, projection-based -kernel construction, and inner/outer approximations with sampling or amplitude estimation for volume, together with Dudley-type spherical-cap packings to prove matching lower bounds. The results provide a complete low-dimensional theory for volume estimation with membership oracles and clarify the potential of quantum and randomized accelerations in this setting, complementing the extensive high-dimensional literature. Overall, the work reveals sharp, dimension-dependent trade-offs: randomness and quantum advantages vanish for convex-set tasks but yield meaningful reductions for volume estimation in fixed dimension.

Abstract

Estimating the volume of a convex body is a canonical problem in theoretical computer science. Its study has led to major advances in randomized algorithms, Markov chain theory, and computational geometry. In particular, determining the query complexity of volume estimation to a membership oracle has been a longstanding open question. Most of the previous work focuses on the high-dimensional limit. In this work, we tightly characterize the deterministic, randomized and quantum query complexity of this problem in the high-precision limit, i.e., when the dimension is constant.

Paper Structure

This paper contains 19 sections, 17 theorems, 24 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Low-dimensional computational geometry.
  3. Our results.
  4. Techniques.
  5. History in the high-dimensional setting.
  6. Organization.
  7. Preliminaries
  8. Notation
  9. Access model and computational models
  10. Rounding convex bodies
  11. Geometry
  12. Problem definitions
  13. Algorithms
  14. Convex set estimation
  15. Volume estimation
  16. ...and 4 more sections

Key Result

Theorem 2.2

Let $d \in \mathbb{N}$, $R \geq 1$, and let $K \subseteq \mathbb{R}^d$ be convex such that $B_d \subseteq K \subseteq RB_d$. Then, there is a deterministic algorithm that makes $O(\mathop{\mathrm{poly}}\nolimits(d,\log(R))$ queries, and finds an invertible affine linear map $L$ such that $B_d \subse

Figures (3)

  • Figure 1.1: Graph of the exponents in the query complexities for the volume estimation problem. $d$ is fixed, and the asymptotic limit is for $R \to \infty$ and $\varepsilon \downarrow 0$. The tilde hides polylogarithmic factors in $1/\varepsilon$ and $R$. The dashed lines represent the previously best-known results, and the solid ones connect the newly-found complexities.
  • Figure 4.1: The lower bound construction in two dimensions with $n = 6$. For any $x \in \{0,1\}^6$, the convex body $K_x$ is formed by taking the union of $K_0$ and all spherical caps $P_j$ if and only if the corresponding bit $x_j$ is $1$.
  • Figure 4.2: The shaded region is a spherical cap around $v$ with radius $r$.

Theorems & Definitions (34)

  • Definition 2.1: Membership oracle
  • Theorem 2.2: grotschel2012geometric
  • proof
  • Definition 2.3: $\delta$-net
  • Proposition 2.4
  • proof
  • Definition 2.5: $\varepsilon$-kernel agarwal2004approximating
  • Lemma 2.6: agarwal2004approximating,agarwal2024computing
  • Lemma 2.7
  • proof
  • ...and 24 more