Affine Spherical Mass and a Gaussian Kinematic Formula for Tropical Two-Fans
Abstract
Let F be an effective balanced rational two-dimensional fan in a rank-four lattice. We introduce an affine spherical mass M*(F)=infcovolg(N)=1Mg(F) and prove the two-sided comparison √(8/3) m(F)≤ M*(F)≤ 3√(2 q(F)). Here q(F)=deg(F· F) is stable self-intersection and m(F) is the least degree of F under a primitive rank-two lattice quotient. For a surjection A, one has deg(A*F)=deg(F·[ker AR]); hence m(F) is equivalently the least stable intersection degree with a complete weight-one rational two-plane. In particular, q(F)≥4/27m(F)2. The proof combines an exact Gaussian--kinematic formula for mixed stable intersection with an affine normalization principle for stationary spherical links. In dimension four, the identification Gr+(2,4)≃ S2× S2 converts ambient isotropy into a sharp second-moment transversality estimate. We prove affine covariance for M*, derive equality and stability conditions for the transversality estimate, and give examples where ambient isotropy coexists with zero mixed intersection, as well as a three-fan in six-space with zero self-intersection. Determining the optimal numerical constant remains open.
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BibTeX
@misc{kalinin2026affine,
title = {{Affine Spherical Mass and a Gaussian Kinematic Formula for Tropical Two-Fans}},
author = {Nikita Kalinin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28498},
eprint = {2609.28498},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28498},
}
RIS
TY - UNPB
TI - Affine Spherical Mass and a Gaussian Kinematic Formula for Tropical Two-Fans
AU - Nikita Kalinin
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.28498
UR - https://arxiv.org/abs/2609.28498
AB - Let F be an effective balanced rational two-dimensional fan in a rank-four lattice. We introduce an affine spherical mass M_*(F)=inf_covol_g(N)=1M_g(F) and prove the two-sided comparison √(8/3) m(F)≤ M_*(F)≤ 3√(2 q(F)). Here q(F)=deg(F· F) is stable self-intersection and m(F) is the least degree of F under a primitive rank-two lattice quotient. For a surjection A, one has deg(A_*F)=deg(F·[ker A_R]); hence m(F) is equivalently the least stable intersection degree with a complete weight-one rational two-plane. In particular, q(F)≥4/27m(F)^2. The proof combines an exact Gaussian--kinematic formula for mixed stable intersection with an affine normalization principle for stationary spherical links. In dimension four, the identification Gr^+(2,4)≃ S^2× S^2 converts ambient isotropy into a sharp second-moment transversality estimate. We prove affine covariance for M_*, derive equality and stability conditions for the transversality estimate, and give examples where ambient isotropy coexists with zero mixed intersection, as well as a three-fan in six-space with zero self-intersection. Determining the optimal numerical constant remains open.
ER -