Abstract
We have set as our goal proving that there is a vm-reduction ordering of the set V of vertices to either side of a quasigeodesic \({\mathcal Q}\) that results in a slit tree Λ, a goal achieved in Chap. 15.
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Notes
- 1.
This is a length version of the area claim in Corollary 9.2.
- 2.
Even focusing on basic facts about convexity, this chapter is the longest in the monograph. Further investigations are left for future work.
- 3.
The “geodesic chain” assumption is not necessary, it only simplifies the discussion.
- 4.
We point out here that a different notion of “quasi-geodesic” also exists, see, e.g., [BZ21].
- 5.
See the discussion in Chap. 18.
- 6.
An extreme point of a convex set S ⊆ P is a point in S that is not interior to any geodesic segment joining two points of S.
- 7.
As we mentioned in the introduction to this chapter, this was known to Archimedes.
- 8.
Of course, the vertices of \({\mathcal Q}\) with α = π are flattened when passing to P#.
- 9.
See Sect. 2.1.
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O’Rourke, J., Vîlcu, C. (2024). Convexity on Convex Polyhedra. In: Resha** Convex Polyhedra. Springer, Cham. https://doi.org/10.1007/978-3-031-47511-5_13
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