What does this research mean for the field?

For any simple polygonal region, the kernel deficit dominates twice the hull deficit, yielding the sharp inequality area(F) · (area(F) − area(K)) ≥ 2 · area(K) · (area(C) − area(F)) and strictly improving Nakano's inequality. Novelty: ClaimNovelty.INCREMENTAL. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

To establish a sharp geometric inequality relating the area of a simple planar polygon, its visibility kernel, and its convex hull, thereby strengthening Nakano's inequality.

August 27, 2026Open Access

The Kernel Deficit Dominates Twice the Hull Deficit: A Sharp Strengthening of Nakano's Inequality

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To establish a sharp geometric inequality relating the area of a simple planar polygon, its visibility kernel, and its convex hull, thereby strengthening Nakano's inequality.
Derived a convex-body cap-union theorem combining a boundary-reversal inequality, support-function identities, and Minkowski's mixed-area inequality.
Extended bounds to all positive-area compact convex bodies via inner polygonal approximation alongside null-area classifications.
Machine-checked the main theorems and proofs using the Lean 4 proof assistant and Mathlib.
Proved the sharp inequality area(F) · (area(F) − area(K)) ≥ 2 · area(K) · (area(C) − area(F)), which translates to G ≤ A / (2 − A) for guard-point ratio G and exterior-area ratio A.
Demonstrated strict improvement over Nakano's inequality (G ≤ A) for every nonconvex polygon.
Confirmed the optimality of the coefficient two through an explicit one-parameter family of nonconvex equality examples.

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Implication

Geometric analysis demonstrates a sharp inequality bounding visibility kernel area relative to convex hull area in simple polygons, strictly improving Nakano's inequality.

Cite This Study

Dakota Charles Baker (2026) studied this question.

synapsesocial.com/papers/6a8fe9b910c91c1e926218fd https://doi.org/10.5281/zenodo.22099578

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Implication

Geometric analysis demonstrates a sharp inequality bounding visibility kernel area relative to convex hull area in simple polygons, strictly improving Nakano's inequality.

The Kernel Deficit Dominates Twice the Hull Deficit: A Sharp Strengthening of Nakano's Inequality

Key Points

Discussion

Implication

Cite This Study

The Kernel Deficit Dominates Twice the Hull Deficit: A Sharp Strengthening of Nakano's Inequality

Key Points

Discussion

Implication

Cite This Study

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Also Consider