On the perimeter, diameter and circumradius of ordinary hyperbolic reduced polygons

TL;DR

This work addresses ordinary reduced polygons in the hyperbolic plane $H^2$ and resolves Lassak's questions on diameter and circumradius, derives a perimeter formula, and establishes a hyperbolic disk-covering property in analogy with Fabińska. It proves a sharp diameter bound $\text{diam}(P)\le 2\,\text{arcosh}\left(\frac{\cosh w+\sqrt{\cosh^2 w+8}}{4}\right)$ with equality only for the regular triangle, and a corresponding circumradius bound $R(P)\le \text{arsinh}\left(\frac{2}{\sqrt{3}}\sqrt{\left(\frac{\cosh w+\sqrt{\cosh^2 w+8}}{4}\right)^2-1}\right)$, again with equality for the regular triangle. It also provides a closed-form perimeter formula $\text{perim}(P)=2\sum_{i=1}^n p_w(\varphi_i)$ and proves that every ordinary reduced polygon of width $w$ is contained in a disk of radius $w$ centered at a boundary point, generalizing Fabińska’s covering result to hyperbolic space. The analysis employs a butterfly decomposition, angle relations, and hyperbolic trigonometry, together with the hyperbolic Jung bound to relate diameter and circumradius. Collectively, the results extend Lassak's Euclidean and spherical findings to $H^2$ and illuminate the distinct extremal behavior of hyperbolic reduced polygons, while highlighting open questions on perimeter extremality and the role of regular polygons in diameter/circumradius minimization.

Abstract

A convex body $R$ in the hyperbolic plane is reduced if any convex body $K\subset R$ has a smaller minimal width than $R$. We answer a few of Lassak's questions about ordinary reduced polygons regarding its perimeter, diameter and circumradius, and we also obtain a hyperbolic extension of a result of Fabińska.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5
• proof
• Lemma 3.6