Holes in 2 - convex point sets ∗
نویسندگان
چکیده
Let S be a finite set of n points in the plane in general position. A k-hole of S is a simple polygon with k vertices from S and no points of S in its interior. A simple polygon P is l-convex if no straight line intersects the interior of P in more than l connected components. Moreover, a point set S is l-convex if there exists an l-convex polygonalization of S. Considering a typical Erdős-Szekeres type problem we show that every 2-convex point set of size n contains a convex hole of size Ω(log n). This is in contrast to the well known fact that there exist general point sets of arbitrary size that do not contain a convex 7-hole. Further, we show that our bound is tight by providing a construction for 2-convex point sets with holes of size at most O(log n).
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