Acute triangulations of polygons. (English) Zbl 1112.52002

Triangulation of a polygon subject to certain condition is a traditional topic in planimetry. The condition can be that all the triangles are acute or non-obtuse. The appropriate triangulations are called respectively acute and non-obtuse. Burago and Zalgaller proved in 1960 that every polygon allows an acute triangulation. Let \(P\) be a class of polygons and \(p \in P\). Denote by \(m(p)\) the minimum size of all acute triangulations of \(p\). Some other results in that area establish for \(P\) upper bounds of \(m(p)\) when \(p \in P\). Say, Machara proved that, if \(P\) is a class of polygons allowing a non-obtuse triangulation of size \(N\), then \(m(p) \leq 2.6^5N\).
The present author improves this result significantly showing that \(m(p) \leq 24N\) (Theorem 2). He shows also that, if \(P\) is the class of all \(n\)-gons, then \(m(p) \leq 106n -216\) (Theorem 3).


52A10 Convex sets in \(2\) dimensions (including convex curves)
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