# Local Complexity of Polygons

Many problems in Discrete and Computational Geometry deal with simple polygons or polygonal regions. Many algorithms and data-structures perform considerably faster, if the underlying polygonal region has low local complexity. One obstacle to make this intuition rigorous, is the lack of a formal definition of local complexity. Here, we give two possible definitions and show how they are related in a combinatorial sense. We say that a polygon P has point visibility width w=pvw, if there is no point q∈ P that sees more than w reflex vertices. We say that a polygon P has chord visibility width w=cvw, if there is no chord c=seg(a,b)⊂ P that sees more than w reflex vertices. We show that cvw ≤ pvw ^O( pvw ), for any simple polygon. Furthermore, we show that there exists a simple polygon with cvw ≥ 2^Ω( pvw ).

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