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Leere konvexe Vielecke in ebenen Punktmengen. (Empty convex polygons in planar point sets). (German) Zbl 0629.52016
Naturwissenschaftliche Fakultät der Technischen Universität Carolo- Wilhelmina Braunschweig. 46 S. (1987).
For any planar n-point set M with no three points collinear, let $$h_ k(M)$$ be the number of empty convex k-gons in M (“empty” means that the convex k-gon has no point of M in its interior). Set $$h_ k(n)=\min \{h_ k(M):| M| =n\}$$. The author shows that $$h_ 3(n)\geq n^ 2-5n+7$$ and $$h_ 4(n)\geq \left( \begin{matrix} n-3\\ 2\end{matrix} \right)$$ and that equalities hold for $$3\leq n\leq 9$$. Moreover, he also shows that $$h_ 3(10)=58$$, $$h_ 4(10)=23$$, $$h_ 5(10)=1$$, $$h_ 3(11)=75$$, $$h_ 4(11)=32$$, $$h_ 5(11)=2$$, $$94\leq h_ 3(12)\leq 95$$, $$42\leq h_ 4(12)\leq 44$$, $$3\leq h_ 5(12)\leq 4$$ and, for all $$n\geq 13$$, $$h_ 3(n)\geq n^ 2- 5n+10,\quad h_ 4(n)\geq \left( \begin{matrix} n-3\\ 2\end{matrix} \right)+6$$ and $$h_ 5(n)\geq 3[n/12]$$.
Reviewer: J.Daneś

##### MSC:
 52A40 Inequalities and extremum problems involving convexity in convex geometry 52A37 Other problems of combinatorial convexity 52A10 Convex sets in $$2$$ dimensions (including convex curves) 52Bxx Polytopes and polyhedra
##### Keywords:
empty convex planar polygon