Exact Finite Integral Geometry: Directional Kernels and Spherical Designs
Abstract
We study when an invariant directional average from integral geometry can be replaced exactly by finitely many directions. For an even continuous kernel ψ:[−1,1]→R, let Tψ be the associated zonal convolution operator on the sphere and let cBψK be the corresponding surface-area observable of a convex body K⊂Rd. Our first result is a sharp norm identity: the worst relative error over all convex bodies equals one half of the L∞ norm of the potential discrepancy Tψ(μ−σ), for every normalized signed directional measure μ. Hence universal exactness is equivalent to μ−σ∈ker Tψ. Funk--Hecke diagonalization then gives a complete spectral criterion: one must annihilate exactly the spherical harmonic degrees on which the multiplier of Tψ is nonzero. This yields kernel-adapted designs and positive finite exact rules for finite active spectrum. For ψp(s)=|s|p we obtain a complete classification: even powers give precisely weighted real projective designs, whereas non-even powers admit no finite signed atomic rule that is exact for every convex body. For the classical Cauchy kernel |s|, exactness fails but spherical t-designs give a uniform O(t−1) relative surface-area error. We also derive exact projection-moment identities for rectifiable submanifolds.
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BibTeX
@misc{an2026exact,
title = {{Exact Finite Integral Geometry: Directional Kernels and Spherical Designs}},
author = {Congpei An},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05869},
eprint = {2609.05869},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05869},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Finite Integral Geometry: Directional Kernels and Spherical Designs
AU - Congpei An
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05869
UR - https://arxiv.org/abs/2609.05869
AB - We study when an invariant directional average from integral geometry can be replaced exactly by finitely many directions. For an even continuous kernel ψ:[−1,1]→R, let T_ψ be the associated zonal convolution operator on the sphere and let cB_ψK be the corresponding surface-area observable of a convex body K⊂R^d. Our first result is a sharp norm identity: the worst relative error over all convex bodies equals one half of the L^∞ norm of the potential discrepancy T_ψ(μ−σ), for every normalized signed directional measure μ. Hence universal exactness is equivalent to μ−σ∈ker T_ψ. Funk--Hecke diagonalization then gives a complete spectral criterion: one must annihilate exactly the spherical harmonic degrees on which the multiplier of T_ψ is nonzero. This yields kernel-adapted designs and positive finite exact rules for finite active spectrum. For ψ_p(s)=|s|^p we obtain a complete classification: even powers give precisely weighted real projective designs, whereas non-even powers admit no finite signed atomic rule that is exact for every convex body. For the classical Cauchy kernel |s|, exactness fails but spherical t-designs give a uniform O(t^−1) relative surface-area error. We also derive exact projection-moment identities for rectifiable submanifolds.
ER -