## 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).

### MSC:

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

### Keywords:

Triangulation; polygon; acute triangulation
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