Sibley's Guard-Point Convexity Measure: A Perimeter Counterexample and a Dominance Bound
2026-06-03 • Computational Geometry
Computational Geometry
AI summaryⓘ
The authors study a way to measure how 'convex' a simple polygon is, called Sibley's guard-point convexity measure, and compare it to two other measures based on the polygon's exterior and perimeter. They prove that Sibley's measure is always less than or equal to the exterior convexity measure. However, they find a specific five-sided shape where Sibley's measure is not always less than the perimeter-based measure. Still, they show a general upper limit holds between these two measures for all polygons. They also introduce a new ratio related to how direction affects perimeter comparisons.
Sibley's guard-point convexitysimple polygonsexterior convexity measureperimeter convexity measureconvexity inequalityanisotropic perimeternonconvex pentagonconvexity bounds
Authors
Masahito Nakano
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.