Exact Finite Integral Geometry: Directional Kernels and Spherical Designs
We study when an invariant directional average from integral geometry can be replaced exactly by finitely many directions. For an even continuous kernel $\psi:[-1,1]\to\R$, let $T_\psi$ be the associated zonal convolution operator on the sphere and let $\cB_\psi K$ be the corresponding surface-area observable of a convex body $K\subset\R^d$. Our first result is a sharp norm identity: the worst relative error over all convex bodies equals one half of the $L^\infty$ norm of the potential discrepancy $T_\psi(\mu-\sigma)$, for every normalized signed directional measure $\mu$. Hence universal exactness is equivalent to $\mu-\sigma\in\ker T_\psi$. Funk--Hecke diagonalization then gives a complete spectral criterion: one must annihilate exactly the spherical harmonic degrees on which the multiplier of $T_\psi$ is nonzero. This yields kernel-adapted designs and positive finite exact rules for finite active spectrum. For $\psi_p(s)=|s|^p$ 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.