On the Stretch Factor of Convex Polyhedra whose Vertices are (Almost) on a Sphere

TL;DR

This paper investigates the stretch factor of convex polyhedra's skeletons when vertices are near or on a sphere, establishing bounds on their spanner properties based on geometric constraints.

Contribution

It proves that skeletons of convex polyhedra with vertices on or near a sphere are spanners with bounds depending on vertex placement and face angles.

Findings

Vertices on the unit sphere yield a (0.999·π)-spanner.

Vertices close to the sphere may not form a spanner.

Bounds depend on radii ratio and face angles.

Abstract

Let $P$ be a convex polyhedron in $\mathbb{R}^3$. The skeleton of $P$ is the graph whose vertices and edges are the vertices and edges of $P$, respectively. We prove that, if these vertices are on the unit-sphere, the skeleton is a $(0.999 \cdot \pi)$-spanner. If the vertices are very close to this sphere, then the skeleton is not necessarily a spanner. For the case when the boundary of $P$ is between two concentric spheres of radii $1$ and $R>1$, and the angles in all faces are at least $\theta$, we prove that the skeleton is a $t$-spanner, where $t$ depends only on $R$ and $\theta$. One of the ingredients in the proof is a tight upper bound on the geometric dilation of a convex cycle that is contained in an annulus.