Sibley's Guard-Point Convexity Measure: A Perimeter Counterexample and a Dominance Bound
For researchers in convexity measures and shape analysis, this resolves an open question about the relationship between guard-point and perimeter convexity, showing the inequality fails pointwise but holds up to a factor of 2.
The paper disproves the pointwise perimeter inequality for Sibley's guard-point convexity measure using a counterexample pentagon with G(F)=62/63 and P(F)=185/189, but proves a uniform bound G(F) ≤ 2P(F) for all simple polygons.
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.