  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 \[ cvw \geq 2^{\Omega( pvw )}.\]
