Volume Estimates for Convex Unions

• The paper introduces randomized approximation schemes, such as FPRAS and NSMC, to estimate the Lebesgue measure of unions of convex sets.
• Algorithmic approaches leverage membership and volume queries, wrapping hull estimators, and SOS techniques to balance complexity and error bounds.
• The methods have significant implications for high-dimensional geometry, combinatorial analysis, and optimization, advancing both theory and practical computation.

A volume estimate for the union of convex sets seeks to determine or sharply bound the Lebesgue measure of $\bigcup_{i=1}^m C_i$ for a finite collection of convex sets $\{C_1,\dots,C_m\}\subset\mathbb{R}^n$. This problem sits at the intersection of high-dimensional geometry, computational complexity, combinatorics, and optimization, with critical applications in stochastic geometry, optimization, computational algebraic geometry, and harmonic analysis. The union operation introduces non-convexity even when the original sets are convex, rendering volume estimation nontrivial both algorithmically and in terms of theoretical structure.

1. Computational Hardness and Oracle Models

Volume computation for unions of convex sets is known to be $\#\mathbf{P}$-hard, subsuming classic problems such as Klee’s measure problem for axis-parallel boxes. The intractability holds even for simple bodies: exact computation of $\mathrm{Vol}(\bigcup_{i=1}^n B_i)$ is $\#\mathbf{P}$-hard even when the $B_i$ are axis-parallel boxes sharing a corner (0809.0835). This intractability motivates relaxation to randomized approximation schemes relying on weak oracles for point membership, uniform sampling, and approximate single-body volume computation:

• POINTQUERY $(x,B)$: Boolean, indicating whether $x\in B'$ (with $B'$ an $\epsilon_P$-approximate body)
• VOLUMEQUERY $(B)$: Returns $V'$ so that $$|(V'-\mathrm{Vol}(B))/\mathrm{Vol}(B)|\le\epsilon_V$$
• SAMPLEQUERY $(B)$: Returns $x\in B'$ almost uniformly, with $f(x)$ close to $1/\mathrm{Vol}(B')$

This oracle model underlies the fastest fully polynomial randomized approximation schemes (FPRAS) and is robust: as long as $\epsilon_P,\epsilon_V,\epsilon_S=O(\epsilon^2/n)$ with $n$ the number of bodies, a $(1\pm\epsilon)$-multiplicative approximation can be obtained (0809.0835).

2. Algorithmic Approaches

2.1 FPRAS for Unions

The APPROXUNION algorithm of Bringmann and Friedrich (0809.0835) constructs a near-optimal FPRAS for the volume of unions of convex bodies. It combines the following steps:

• Compute approximate volumes $V_i$ for each $B_i$.
• Perform a randomized "hit-and-judge" loop: select $B_i$ with probability proportional to $V_i$, draw $x\sim B_i$, and determine the number of random $B_j$ needed for $x$ to be classified as contained.
• Return the estimator $\widetilde{U} = \frac{T_\text{sum}\, V'}{n\,M}$ where $V' = \sum_i V_i$.

When convex bodies are provided by membership oracles, and volume/sampling oracles are simulated using Dyer–Frieze–Kannan or Lovász–Vempala random walks, the FPRAS has total runtime $\widetilde O(nd^5/\epsilon^2)$ for $d$-dimensional bodies (0809.0835).

2.2 $n$-Sphere Monte Carlo (NSMC)

The $n$-sphere Monte Carlo method, due to Venkatapathi and Arun, provides a volume estimator by sampling random directions $\hat{s}\in S^{n-1}$ and measuring the maximal extent function $S(\hat{s}) := \sup\{\rho: \rho\hat{s} \in D\}$ for the set $D$ containing the origin. For unions of convex sets:

• $S(\hat{s}) = \max_{1\leq i\leq m} S_i(\hat{s})$ where $S_i$ is the support function of $C_i$.
• The volume is $V(D) = v_n\, \mathbb{E}_{\hat{s}}[S(\hat{s})^n]$ where $v_n$ is the volume of the unit $n$-ball.
• Complexity is $O(m n^2/\epsilon^2)$ for relative error $\epsilon$, with provable error bounds (I. et al., 2020).
• NSMC is dimension-agnostic, “embarrassingly parallel,” and outperforms multistage MCMC methods for $n\lesssim 100$ with moderate accuracy requirements.

2.3 Wrapping Hull Estimation

The wrapping hull approach of Baldin (Baldin, 2017) provides a statistically optimal estimator for the volume of a union of convex sets, within the intersection-stable class. Given $N$ points sampled uniformly from $S = \bigcup_{i=1}^m C_i$, define the wrapping hull $W$ as the minimal union of $\leq m$ convex bodies containing the sample:

• Oracle (known $\lambda$): $$\widehat{V}_\text{oracle} = |W| + N_\partial/\lambda$$ where $N_\partial$ is the number of sample points on $\partial W$.
• Data-driven: $\widehat{V} = \frac{N+1}{N_0+1}|W|$ with $N_0$ interior sample points.
• The estimator achieves minimax error rates $O(n^{-2/(d+1)})$ in dimension $d$.

This framework is unbiased, consistent, and attains the uniformly minimal variance among all estimators based on the observed samples (Baldin, 2017).

2.4 Sum-of-Squares (SOS) Semialgebraic Methods

SOS-based methods enable inner and outer semialgebraic approximations of unions of convex sets. Given $S = \bigcup_{i=1}^m C_i$, one can encode the inclusion $S \subseteq \{x : p(x)\leq 1\}$ for a degree-$2d$ polynomial $p(x)$ via SOS certificates:

• For each $C_i = \{x : g_{i,j}(x)\geq 0\,\forall j\}$, enforce $$p(x)-1 = \sigma_i(x) + \sum_j \lambda_{i,j}(x)g_{i,j}(x)$$ with $\lambda_{i,j},\sigma_i \in \Sigma_\mathrm{SOS}$.
• Add convexity constraints if required: enforce the Hessian $\nabla^2 p(x)\succeq 0$ via matrix-SOS.
• Minimize $-\log\det(Q)$ where $p(x)=v(x)^T Q v(x)$ to heuristically tighten volume bounds (Jones et al., 2018).

Volume is then estimated for the 1-sublevel set by randomized or moment methods. This approach is heuristic for $d>2$, but yields tight convex relaxations especially in the quadratic case.

3. Mixed Volume, Polytope Unions, and the “Unmixing” Principle

The computational algebraic geometry of unions is informed by the concept of mixed volume. "Unmixing the mixed volume" refers to sufficient combinatorial face conditions on the family $\{P_i\}$ of convex polytopes guaranteeing:

$\MV(P_1, \ldots, P_n) = n!\operatorname{Vol}\big(\operatorname{conv}(P_1 \cup \ldots \cup P_n)\big)$

(Chen, 2017). The “unmixed-faces” and “semi-mixed-faces” theorems give explicit criteria: if every positive-dimensional face of the convex hull meets every $P_i$ (or, for semi-mixed, more weakly intersects in low-dimensional faces in a controlled way), exact reduction is possible.

Algorithmic implication: traditional mixed volume routines scale poorly with the Cayley complexity, but replacing the computation with a single convex hull typically leads to speedups of one or two orders of magnitude, as demonstrated in power-flow benchmarks and polyhedral homotopy root-counting (Chen, 2017). For general convex bodies an analogous extension remains conjectural.

4. Combinatorial and Geometric Volume Lower Bounds

Recent advances relate the structure of unions to combinatorial volume lower bounds. If a family $\mathcal{U}$ of convex sets (of diameter $\leq 1$ and volume $\sim \delta^2$ in $\mathbb{R}^3$) satisfies a "no clustering in convex sets" condition, sharp lower volume bounds follow:

$$\left| \bigcup_{U\in \mathcal{U}} U \right| \geq c \,\delta^\epsilon N^{-1} (\#\mathcal{U})\,\delta^2$$

where $N$ is such that no convex $V$ contains more than $N$ elements of $\mathcal{U}$ (Wang et al., 24 Feb 2025). This generalizes classical Kakeya-type estimates and is proved by multiscale induction, grains decomposition, and hairbrush bounds. The approach holds for both tubes and general convex sets, with applications to the Kakeya conjecture and dimension results.

5. Error Bounds, Complexity, and Practical Guidance

Method	Error (relative/absolute)	Complexity	Applicability
FPRAS (APPROXUNION)	$1\pm\epsilon$ multiplicative	$\widetilde{O}(nd^5/\epsilon^2)$	High-dim, generic
NSMC	$O(\sqrt{n}/\sqrt{N})$ RMS	$O(m n^2/\epsilon^2)$	$n\lesssim100$
Wrapping hull	$O(n^{-2/(d+1)})$ RMS	$O(n^{O(dm)})$ (low-d $m$)	Sample-based, union
SOS (semialgebraic outer)	Heuristic, sample/analytic	Depends on SDP size	Convex polynomial input

In practice, the NSMC is favorable for moderate dimension if the support functions are explicit, parallellizable, and unbiased. FPRAS works widely when oracles are accessible. Wrapping hull estimators achieve minimax error rates when uniform samples are available. SOS approaches are best suited where tight outer convex or semialgebraic approximations are needed (I. et al., 2020, 0809.0835, Baldin, 2017, Jones et al., 2018).

6. Theoretical Implications and Open Questions

The study of volume estimation for unions of convex sets has led to important insights into complexity, geometric measure theory, and computational algebra. Notably:

• There is a provable asymmetry: unions of convex sets admit efficient randomized approximation, but intersections are provably hard to approximate multiplicatively in high-dimensions (0809.0835).
• The connection between mixed volume and the convex hull of unions allows practical speed-ups for algebraic geometry applications under combinatorial criteria (Chen, 2017).
• Geometric lower bounds for unions with anti-concentration structure yield sharp estimates in combinatorial and harmonic analysis contexts, resolving dimension conjectures (Wang et al., 24 Feb 2025).

A plausible implication is that advances in volume estimates for unions may further impact the study of exceptional sets in geometric measure theory and the analysis of polynomial systems, with direct computational and theoretical utility. Extensions from polytopes to general convex bodies in the mixed-volume equivalence, and optimizing the computational cost for large-scale unions given only oracle access, remain key open directions.