A new mathematics preprint reports a universal factor-of-two relationship between two ways of measuring how far a simple polygonal region falls short of simpler shapes. The theorem says the area lost when a polygon is reduced to its visibility kernel is at least twice the kernel-weighted area gap between the polygon and its convex hull.
In plain terms, the visibility kernel is the part of an irregular polygon from which the whole region can be seen, while the convex hull is the tightest convex shape containing it. The result links those two geometric deficits for every simple polygonal region, rather than only for a selected collection of examples.
A stricter bound on guard points
The same theorem can be written as a bound on a guard-point ratio: G(F) is no greater than A(F)/(2−A(F)). The paper says this improves Nakano's earlier inequality, G(F) ≤ A(F), whenever A(F) is below 1.
These are geometric quantities associated with the polygon, not measurements from an experiment. There is no statistical uncertainty here: the result is a deterministic theorem about mathematical objects, including polygonal regions, compact convex sets and finite point sets.
The proof reduces the problem to a convex-geometric estimate
The proof combines a sorting of polygon edges around the boundary with a reversal of boundary edges, then joins that rearrangement with two area estimates. One is a companion-area bound, and the other uses support functions, which record how far a shape extends in each direction, together with mixed-area inequalities.
Its central auxiliary result is a cap-union inequality. For a nonempty compact convex set K and a finite nonempty point set X with K contained in the convex hull of X, the union U of the relevant caps satisfies |U|² + |K||U| ≥ 2|K||C|, where C is the associated convex hull. The statement does not require K to have positive area and does not impose extra conditions such as convex position or an irredundant point set.
To extend the argument to arbitrary compact convex K, the proof uses an inner polygonal approximation that keeps a fixed disk inside K. This route avoids relying on area continuity for arbitrary nonconvex unions.
Why the number two matters
The coefficient is not presented as a convenient rounding point. The paper gives a nonconvex equality family on which replacing 2 with any larger universal coefficient fails. That establishes sharpness of the factor, although the manuscript does not claim to classify every equality case.
A formal check, with limits
Both main theorems have machine-checked Lean 4 proofs, and the final formal statements were audited against the informal statements after kernel checking. The completed version also reports passed direct compilation and passed package-target, research-aggregate and full builds, with no sorryAx or project-specific axiom in the public theorem dependency closures.
The document is a preprint, identified as Preprint v6 and arXiv:2608.25057v1. Its separate referee and citation-verification passes were automated checks rather than human peer review, so the formal build record should not be confused with journal review.
The author discloses using Claude and ChatGPT/Codex for exploratory mathematics, Lean formalization, manuscript organization, citation checking and language editing, while accepting responsibility for the manuscript.
A related optimization is still unsettled
The preprint also discusses a separate perimeter program for an optimum called α∗. That program reports a certified interval of 1.06107423573065456901 < α∗ ≤ 2, based on an explicit 503-gon certificate and Nakano's theorem. This is a certified lower and upper bound, not an exact determination.
A proposed comb-family limit of 1.0614097453924 is explicitly not asserted to equal α∗. Global optimality of the combs, convergence to a continuum problem and the limiting value remain unproved.
The proved results also stop short of a complete equality classification, a quantitative stability theorem, higher-dimensional analogues or an extension to arbitrary compact generator sets. The paper presents those questions, along with a rigorous solution of the proposed continuum comb optimization problem, as open directions rather than settled conclusions.
The numerical material is packaged in a supplement distributed through a stable Zenodo concept record, with exact certificate data and a minimal verifier. The result is a mathematical claim, not empirical evidence about people, animals or clinical outcomes.
Paper data and sources
Original title: The Kernel Deficit Dominates Twice the Hull Deficit: A Sharp Strengthening of Nakano's Inequality
Authors: Dakota Charles Baker
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text