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The common exterior of convex polygons in the plane. (English) Zbl 0881.68122
Summary: We establish several combinatorial bounds on the complexity (number of vertices and edges) of the complement of the union (also known as the common exterior) of $$k$$ convex polygons in the plane, with a total of $$n$$ edges. We show: (1) The maximum complexity of the entire common exterior is $$\Theta(n\alpha(k)+ k^2)$$. (2) The maximum complexity of a single cell of the common exterior is $$\Theta(n\alpha(k))$$. (3) The complexity of $$m$$ distinct cells in the common exterior is $$O(m^{2/3}k^{2/3}\log^{1/3}(k^2/m)+ n\log k)$$ and can be $$\Omega(m^{2/3}k^{2/3}+ n\alpha(k))$$ in the worst case.

##### MSC:
 68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
##### Keywords:
combinatorial complexity; convex polygons; common exterior
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##### References:
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