Maximal Distortion of Geodesic Diameters in Polygonal Domains

TL;DR

This work investigates the maximal distortion between geodesic and Euclidean diameters in convex polygonal domains with convex holes, defining $g(h)=\sup_{P\in\mathcal{C}(h)} \varrho(P)$ and proving $\Omega(h^{1/3})\le g(h)\le O(h^{1/2})$. It refines the bounds when hole diameters are bounded by $\Delta$ or holes are fat, yielding $\varrho(P)\le O(1+\min\{h^{3/4}\Delta, h^{1/2}\Delta^{1/2}\})$ and $\varrho(P)=O(1)$ for fat holes. The paper also connects polygonal-domain distortion to triangulation distortion, showing $g(h)=\Theta(f(n))$ where $f(n)=\sup_{G\in\mathcal{T}(n)} \varrho(G)$, thus aligning growth rates with $n$-vertex triangulations. It further discusses an escape-problem formulation and analyzes special hole families (e.g., axis-aligned rectangles, fat holes), highlighting implications for geometric routing and algorithmic computation of geodesic diameters. Open questions remain on tightening the bounds and extending results to higher dimensions.

Abstract

For a polygon $P$ with holes in the plane, we denote by $\varrho(P)$ the ratio between the geodesic and the Euclidean diameters of $P$. It is shown that over all convex polygons with $h$~convex holes, the supremum of $\varrho(P)$ is between $Ω(h^{1/3})$ and $O(h^{1/2})$. The upper bound improves to $\varrho(P)\leq O(1+\min\{h^{3/4}Δ,h^{1/2}Δ^{1/2}\})$ if the Euclidean diameter of every hole is most $Δ$ times the Euclidean diameter of $P$; and to $O(1)$ if every hole is a \emph{fat} convex polygon. Furthermore, we show that the function $g(h)=\sup_P \varrho(P)$ over convex polygons with $h$ convex holes has the same growth rate as an analogous quantity over geometric triangulations with $h$ vertices when $h\rightarrow \infty$.

Figures (5)

• Figure 1: Left: hexagons $Q_0, \ldots, Q_3$ for $k=6$. Right: The 18 holes corresponding to the edges of $Q_1,\ldots , Q_3$.
• Figure 2: Left: A polygon $P\in \mathcal{C}(7)$ with 7 convex holes, a point $s\in P$, and a path ${\rm greedy}_P(s,\vec{u})$ from $s$ to a point $t$ on the outer boundary of $P$. Right: A boundary arc $\widehat{pq}$, where $|\widehat{pq}|\leq |pr|+|rq|$.
• Figure 3: Left: An example, where ${\rm greedy}_P(s,\vec{u})$ visits $P_5$ twice. Right: An example, where the bound $|{\rm greedy}_P(s,\vec{u})|\leq O(h^{1/2})\cdot {\rm diam}_2(P)$ is attained for all $\vec{u}\in \mathbb{S}^1$.
• Figure 4: Illustration for $\ell=3$. Left: All lines $H_i$ and $V_i$. Right: Box $B$ and a path $\gamma$ from $s$ to the outer boundary.
• Figure 5: Line $L$ traverses a convex polygon $P$, but does not cross the segment $ab$.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Proposition 1
• Proposition 2
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1