🤖 AI Summary
This study investigates the pointwise relationship between Sibley’s guard-point convexity measure \( G(F) \) and the perimeter-based convexity measure \( P(F) \). Through geometric construction and inequality analysis, the authors present the first counterexample—a non-convex pentagon—for which \( G(F) = 62/63 > 185/189 = P(F) \), thereby disproving the conjecture that \( G(F) \leq P(F) \) holds pointwise. Concurrently, they establish a universal upper bound \( G(F) \leq 2P(F) \), demonstrating that the two measures remain asymptotically non-dominating. This work clarifies the theoretical boundaries between these two convexity measures and strengthens the mathematical foundation for comparing convexity in simple polygons.
📝 Abstract
We study Sibley's guard-point convexity measure for simple polygons and compare it with the exterior and perimeter convexity measures. We prove the exterior inequality G(F) <= E(F) and disprove the pointwise perimeter inequality G(F) <= P(F) by an explicit nonconvex pentagon with G(F) = 62/63 and P(F) = 185/189. Nevertheless, we prove the uniform bound G(F) <= 2P(F) for every simple polygon. Thus the pointwise perimeter inequality is false, but the corresponding asymptotic non-domination conclusion remains true. We also record an auxiliary guard-point-adapted anisotropic perimeter ratio, which isolates the directional loss in the Euclidean perimeter comparison.