Numerical cubature from Archimedes' hat-box theorem

TL;DR

This work extends Archimedes' hat-box theorem through toric moment maps to develop a broad program for numerical cubature: projection, fibration, and torus-based constructions yield efficient high-dimensional cubature formulas on spheres, simplices, and projective spaces, with concrete results such as a $7$-cubature on $S^n$ using $O(n^4)$ points and $O(n^t)$-point trigonometric cubatures on tori. It provides both constructive methods and sharp lower bounds, including Stroud-type algebraic bounds and local density estimates on the simplex that force dispersion of points and weights. By linking cubature to lattice packing, Hadamard designs, and the geometry of toric varieties, the paper unifies several known designs (e.g., Simpson's rule, Gauss-type rules) and yields new near-optimal configurations, advancing theory and potential applications in high-dimensional numerical integration. The results illuminate the trade-offs between design strength, positivity/interiority of weights, and the number of evaluation points, with Gaussian quadrature shown to be highly locally optimal among positive cubatures. The work also opens avenues for further exploration of Fourier and lattice-based cubature limits and connections to algebraic geometry and combinatorial designs.

Abstract

Archimedes' hat-box theorem states that uniform measure on a sphere projects to uniform measure on an interval. This fact can be used to derive Simpson's rule. We present various constructions of, and lower bounds for, numerical cubature formulas using moment maps as a generalization of Archimedes' theorem. We realize some well-known cubature formulas on simplices as projections of spherical designs. We combine cubature formulas on simplices and tori to make new formulas on spheres. In particular $S^n$ admits a 7-cubature formula (sometimes a 7-design) with $O(n^4)$ points. We establish a local lower bound on the density of a PI cubature formula on a simplex using the moment map. Along the way we establish other quadrature and cubature results of independent interest. For each $t$, we construct a lattice trigonometric $(2t+1)$-cubature formula in $n$ dimensions with $O(n^t)$ points. We derive a variant of the Möller lower bound using vector bundles. And we show that Gaussian quadrature is very sharply locally optimal among positive quadrature formulas.

Figures (10)

• Figure 1: .
• Figure 2: .
• Figure 3: .
• Figure 4: .
• Figure 5: .

Theorems & Definitions (34)

• Proposition 3.1
• Theorem 3.2
• Theorem 3.3
• Remark
• proof
• Remark
• Theorem 4.1
• proof
• Remark
• Corollary 4.2