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Positive cubature on S2:low-degree rigidity and uniform bounds

  • Zhuo Cheng
  • Deyu Yu
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Abstract

Let Nt denote the least number of nodes in a positive cubature formula of degree t on S2. We prove that a formula of degree 2m+1 cannot have exactly (m+1)(m+2)+1 nodes for any m≥2. Excluding equality in the Fisher bound then gives N2m+1≥ (m+1)(m+2)+2 for m≥3, and known constructions yield the exact values N7=22 and N9=32. For general odd degrees, positive circle measures on Lobatto latitudes give an upper bound with quadratic coefficient 13/8, parity-dependent linear terms, and an O(m2/3) remainder. We determine the sharp constant 49√[3]3/72 for the scalar remainder in this construction. Lower bounds are obtained from continuous weighted LP--Turán inequalities and radial caps whose Helmholtz companions are nonnegative measures. We prove that the cap functional admits a maximizer at each fixed admissible support radius and derive explicit finite-degree lower bounds by a positivity-preserving transfer to the sphere.

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Cite

BibTeXRIS
BibTeX
@misc{cheng2026positive,
  title = {{Positive cubature on $S^2$:low-degree rigidity and uniform bounds}},
  author = {Zhuo Cheng and Deyu Yu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.40210},
  eprint = {2609.40210},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40210},
}
RIS
TY  - UNPB
TI  - Positive cubature on S^2:low-degree rigidity and uniform bounds
AU  - Zhuo Cheng
AU  - Deyu Yu
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.40210
UR  - https://arxiv.org/abs/2609.40210
AB  - Let N_t denote the least number of nodes in a positive cubature formula of degree t on S^2. We prove that a formula of degree 2m+1 cannot have exactly (m+1)(m+2)+1 nodes for any m≥2. Excluding equality in the Fisher bound then gives N_2m+1≥ (m+1)(m+2)+2 for m≥3, and known constructions yield the exact values N_7=22 and N_9=32. For general odd degrees, positive circle measures on Lobatto latitudes give an upper bound with quadratic coefficient 13/8, parity-dependent linear terms, and an O(m^2/3) remainder. We determine the sharp constant 49√[3]3/72 for the scalar remainder in this construction. Lower bounds are obtained from continuous weighted LP--Turán inequalities and radial caps whose Helmholtz companions are nonnegative measures. We prove that the cap functional admits a maximizer at each fixed admissible support radius and derive explicit finite-degree lower bounds by a positivity-preserving transfer to the sphere.
ER  -