Dual Geometry of Spherical Designs: Polarity, Self-Polar Rigidity, and Quadrature Structure
Abstract
Let X={x1,…,xN}⊂Sphd−1 be a spherical t-design, t≥2, and let PX⊂Rd be its Minkowski polytope. If hi=hPX(xi), then PX∘=conv{xi/hi:1≤ i≤ N}, so the design is the radial projection of the polar vertex set. This yields exact degree-dependent moment identities and quantitative control of the unweighted polar moments from the Hausdorff sphericity of PX. Our main results concern self-polarity. If PX∘=cUPX, U∈ O(d), we obtain the structured slack factorization A=c hhT−XTUTX, rank A=d+1, whose zero pattern records the facet--vertex incidences. For node-transitive designs, the common incidence level equals the inradius-to-circumradius ratio r/R. Combining this with the one-dimensional moment problem underlying the Fazekas--Levenshtein covering bound, we prove r/R≥ηt,d, with equality forcing the twisted inner-product rows to realize the corresponding Gaussian or Gauss--Radau quadrature rule; in particular, the quantities Nλk become integer incidence multiplicities. We further prove a quantitative near-equality theorem: if δ=r/R−ηt,d is small, then each row is Od,t(√δ)-close in W1 to the extremal quadrature measure, yielding near-incidence rigidity and an arithmetic stability gap. As applications, we obtain three-dimensional rigidity and identify the regular simplex and the $24$-cell as the regular self-polar examples.
Related
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- Gauss--Chebyshev Quadrature, 𝕊1-Designs, and the Planar Minkowski Inverse Problem: A One-Dimensional Model for Design-Induced Convex Geometry
- Smallest quotients and profinite rigidity of irreducible spherical type Artin groups
- Sharp local scalar-curvature rigidity of spherical caps
BibTeX
@misc{an2026dual,
title = {{Dual Geometry of Spherical Designs: Polarity, Self-Polar Rigidity, and Quadrature Structure}},
author = {Congpei An},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02439},
eprint = {2609.02439},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02439},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dual Geometry of Spherical Designs: Polarity, Self-Polar Rigidity, and Quadrature Structure
AU - Congpei An
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02439
UR - https://arxiv.org/abs/2609.02439
AB - Let X={x_1,…,x_N}⊂Sph^d−1 be a spherical t-design, t≥2, and let P_X⊂R^d be its Minkowski polytope. If h_i=h_P_X(x_i), then P_X^∘=conv{x_i/h_i:1≤ i≤ N}, so the design is the radial projection of the polar vertex set. This yields exact degree-dependent moment identities and quantitative control of the unweighted polar moments from the Hausdorff sphericity of P_X. Our main results concern self-polarity. If P_X^∘=cUP_X, U∈ O(d), we obtain the structured slack factorization A=c hh^T−X^TU^TX, rank A=d+1, whose zero pattern records the facet--vertex incidences. For node-transitive designs, the common incidence level equals the inradius-to-circumradius ratio r/R. Combining this with the one-dimensional moment problem underlying the Fazekas--Levenshtein covering bound, we prove r/R≥η_t,d, with equality forcing the twisted inner-product rows to realize the corresponding Gaussian or Gauss--Radau quadrature rule; in particular, the quantities Nλ_k become integer incidence multiplicities. We further prove a quantitative near-equality theorem: if δ=r/R−η_t,d is small, then each row is O_d,t(√δ)-close in W_1 to the extremal quadrature measure, yielding near-incidence rigidity and an arithmetic stability gap. As applications, we obtain three-dimensional rigidity and identify the regular simplex and the $24$-cell as the regular self-polar examples.
ER -