On the stretch factor of convex polyhedra whose vertices are (almost) on a sphere

Michiel Smid, Prosenjit Bose, Paz Carmi, Mirela Damian, Jean-Lou De Carufel, Darryl Hill, Anil Maheshwari, Yuyang Liu


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.



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DOI: http://dx.doi.org/10.20382/jocg.v7i1a19

ISSN: 1920-180X