Discrete geometry
Preprint
Exact coordinate-balancing calculus and sharp stability for symmetric lattice cross-covariograms
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- Preprint
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- v1 · Zenodo
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Abstract
For finite \(A, B \subset \mathbb{Z}^d\), write \(g_{A,B}(u) = |A \cap (B - u)|\). We study how this cross-covariogram varies as the coordinates of \(u\) are balanced along a fixed \(\ell_1\) shell. A centered exchange-fiber property, satisfied by lattice sections of compact convex sets invariant under signed coordinate permutations, decomposes every unit balancing transfer into parity-fiber endpoint exposures in \(\{0, 1\}\), yielding the Schur order on each shell and a fiberwise equality certificate for comparable shifts. For equal Lee balls of radius \(t\) in dimension \(d \ge 3\) and comparable shifts on an active shell \(2 \le s \le 2t\), the overlap deficit is at least a closed-form constant times the number of unit transfers; a central-box section theorem evaluates it in lower-dimensional Lee-ball counts. Incomparable shifts may share a value, so equality is order-theoretic. The extremizers are characterized for \(4 \le s \le 2t\): each interior shell carries exactly \(\lfloor s/2 \rfloor - 1\) balanced-gap minimizing orbits, independently of \(d\) and \(t\), with equality only across single such transfers; the two outermost shells form a boundary family. A structural theorem evaluates the sharp layer in every dimension for the Minkowski interpolants \(C_d + bB^d_\infty\) between cross-polytope and cube: the constant is determined by lattice-point counts of the same body two dimensions down, plus an explicit facet surcharge for transfers into a zero coordinate, and one inequality decides the minimizing transfer. Cubes and Lee balls are its degenerate regimes. Convex hyperoctahedral symmetry thus fixes the transfer sign but neither the sharp scale nor the minimizing orbit; reversals occur once convexity or coordinate-permutation symmetry fails.
Supplementary appendix (PDF) · Cryptographic supplementary material.
Keywords
MSC — 52C07 · 11H06 · 52B20 · 05A20
Key contributions
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Universal parity-exact transfer calculus Theorem 3.4
A centered exchange fiber has an explicit endpoint exposure in \(\{0, 1\}\). Summing these exposures proves fixed-shell majorization and a fiberwise equality certificate for any pair of hyperoctahedrally invariant compact convex lattice sections.
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Sharp Lee stability, classification, and radial equality Section 4
For unequal Lee balls the full two-parity kernel and residual convolution are derived. For equal balls in dimension \(d \ge 3\), a central-box theorem evaluates the exact active-shell stability constant by an even–odd formula in lower-dimensional Lee-ball counts; a classification theorem lists all minimizing arcs for \(4 \le s \le 2t\) — on interior shells exactly the \(\lfloor s/2 \rfloor - 1\) balanced-gap axial transfers, a count independent of \(d \ge 3\) and \(t\) — and a rigidity theorem characterizes all pairs attaining equality in the stability inequality, confining them on interior shells to single arcs.
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A structural theorem with an exact facet threshold Theorem 6.4
For the interpolating family \(C_d + bB^d_\infty\) in every dimension, a master kernel reduces each arc weight to a lower-dimensional lattice-point count plus an explicit facet surcharge carried only by transfers into a zero coordinate. This evaluates the sharp constant and the minimizing orbit on every shell below the boundary layer, and locates the exact threshold \(\Delta_{d,b}(s) = 0\) at which the minimizing orbit changes. Integer cubes and equal Lee balls are the two degenerate regimes of the same mechanism, with, respectively, one and \(\lfloor s/2 \rfloor - 1\) minimizing arcs. Capped cross-polytopes add an exact magnitude crossover, and counterexamples exhibit sign reversals once convexity or coordinate-permutation symmetry is removed from the sufficient hypothesis.
Cite this preprint
BibTeX
@misc{lu2026covariogram,
author = {Shunqi Lu},
title = {Exact coordinate-balancing calculus and sharp stability
for symmetric lattice cross-covariograms},
year = {2026},
version = {v1},
publisher = {Zenodo},
doi = {10.5281/zenodo.22178367},
url = {https://doi.org/10.5281/zenodo.22178367}
}Shunqi Lu. (2026). Exact Coordinate-Balancing Calculus and Sharp Stability for Symmetric Lattice Cross-Covariograms (v1). Zenodo. https://doi.org/10.5281/zenodo.22178367