Sibley's Guard-Point Convexity Measure: A Perimeter Counterexample and a Dominance Bound

Abstract

For a simple polygonal region $F$, let $K(F)$ be its polygon kernel, or Sibley's guard-point set, and let $C=\mathrm{conv}(F)$. The associated guard-point, exterior, and perimeter measures are $G(F)=|K(F)|/|F|$, $E(F)=|F|/|C|$, and $P(F)=\mathrm{Per}(C)/\mathrm{Per}(F)$. Using a kernel-adapted anisotropic perimeter, we prove $G(F)\le E(F)$. We disprove the pointwise inequality $G(F)\le P(F)$ by an explicit nonconvex pentagon with integer coordinates for which $G(F)=62/63$ and $P(F)=185/189$. Nevertheless, $G(F)\le 2P(F)$ holds for every simple polygon, and hence $G$ cannot asymptotically dominate $P$ in Sibley's sense. Thus the two assertions in Sibley's Conjecture 2 are settled in opposite directions. We also observe that Sibley's convexity coefficient $\chi$ and interior measure $I$ coincide, respectively, with the Beer index $b$ and convexity ratio $c$. The theorem $b\le 180c$ of Balko, Jelínek, Valtr, and Walczak therefore yields $\chi \le 180I$, resolving Sibley's Conjecture 1.

15 pages, 2 figures. A conference abstract based on v1 was accepted for presentation at JCDCG^3 2026. v2: added the observation that Sibley's Conjecture 1 follows from an earlier theorem on the Beer index; added affine-covariance results, a second figure, and self-contained appendices; revised and expanded the exposition. The main results on Conjecture 2 are unchanged