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 $q\in P$ that sees more than $w$ reflex vertices. We say that a polygon $P$ has chord visibility width $w=cvw$, if there is no chord $c=\textrm{seg}(a,b)\subset P$ that sees more than w reflex vertices. We show that $$cvw \leq pvw ^{O( pvw )},$$ for any simple polygon. Furthermore, we show that there exists a simple polygon with $$cvw \geq 2^{\Omega( pvw )}.$$