Geodesic cycles on the Sphere: $t$-designs and Marcinkiewicz-Zygmund Inequalities

TL;DR

This work studies geodesic cycles on the sphere as both $t$-design curves and carriers of Marcinkiewicz–Zygmund inequalities. It develops a two-track approach: explicit analytic and numerically beautified geodesic cycles that realize $t$-design properties for small $t$ (notably $t=2$ and $t=3$ on $\mathbb{S}^2$), and a general existence theory showing curves achieving mz inequalities with length growth $\ell(\gamma^{(t)}_{d,\varepsilon}) \le C_{d,\varepsilon} t^{d-1}$ in any dimension. The construction combines a two-step numerical minimization and a rigorous beautification procedure, including parameterized families derived from Platonic solids and symmetry arguments, to produce $2$- and $3$-design geodesic cycles and guided attempts toward $5$-design cycles. Separately, the paper builds a mz-curve framework via area-regular partitions and Euler cycles, establishing that curves crossing $t$-design partitions inherit norm equivalences $A_d^{1/p}(1-\varepsilon)\|f\|_{L^p} \le \|f\|_{L^p(\gamma)} \le B_d^{1/p}(1+\varepsilon)\|f\|_{L^p}$ with length $O(t^{d-1})$, thus enabling discretization of $L^p$ norms along curves. The results advance curve-based sampling and quadrature on spheres, with potential implications for mobile sampling and curve-aware numerical integration on spherical domains.

Abstract

A geodesic cycle is a closed curve that connects finitely many points along geodesics. We study geodesic cycles on the sphere in regard to their role in equal-weight quadrature rules and approximation.

Figures (6)

• Figure 1: Visualizations of the geodesic $t$-design cycles constructed in Section \ref{['sec:num']}.
• Figure 2: Curves in Example \ref{['ex:S2']}.
• Figure 4: Numerical candidates of geodesic $5$-design cycles.
• Figure 5: The geodesic chains $\Gamma^{(2,a)}$ and $\Gamma^{(3,a)}$ in Theorem \ref{['thm:2dt simple']}.
• Figure 6: Each of the functions $h_2$ and $h_3$ in \ref{['eq:a2 parameter']} and \ref{['eq:a3 parameter']} has exactly one root in the interval $(0,1]$.

Theorems & Definitions (22)

• Theorem 1.1
• Example 2.1: Smooth $t$-design curves in $\mathbb{S}^2$
• Proposition 2.1
• proof
• Example 2.2: Smooth $t$-design curves in $\mathbb{S}^{2m-1}$
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Remark 3.1