On the Doubling Dimension and the Perimeter of Geodesically Convex Sets in Fat Polygons

TL;DR

This paper investigates the doubling dimension of fat polygons and the perimeter of geodesically convex sets within them, providing bounds and algorithms for computational problems in these geometric spaces.

Contribution

It establishes that certain classes of fat polygons have bounded doubling dimension and derives new perimeter bounds, enabling efficient algorithms for geometric problems.

Findings

Locally-fat simple polygons do not always have bounded doubling dimension.

$(oldsymbol{ ext{α,β}})$-covered polygons have bounded doubling dimension, even with holes.

Perimeter of geodesically convex sets in $(oldsymbol{ ext{α,β}})$-covered polygons is proportional to Euclidean diameter.

Abstract

Many algorithmic problems can be solved (almost) as efficiently in metric spaces of bounded doubling dimension as in Euclidean space. Unfortunately, the metric space defined by points in a simple polygon equipped with the geodesic distance does not necessarily have bounded doubling dimension. We therefore study the doubling dimension of fat polygons, for two well-known fatness definitions. We prove that locally-fat simple polygons do not always have bounded doubling dimension, while any $(\alpha,\beta)$-covered polygon does have bounded doubling dimension (even if it has holes). We also study the perimeter of geodesically convex sets in $(\alpha,\beta)$-covered polygons (possibly with holes), and show that this perimeter is at most a constant times the Euclidean diameter of the set. Using these two results, we obtain new results for several problems on $(\alpha,\beta)$-covered polygons, including an algorithm that computes the closest pair of a set of $m$ points in an $(\alpha,\beta)$-covered polygon with $n$ vertices that runs in $O(n + m\log{n})$ expected time.