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Global Geometry of Polygons. 1: The Theorem of Fabricius-Bjerre

by Thomas F. Banchoff

Suppose X:[a,b] -> E^2 is a closed curve with C crossings and
F inflection points (or inflection intervals), where C and F
are finite. Furthermore, assume that X has finitely many double
support lines. A support line is a line through two points of
a curve such that each point has a neighborhood lying on one side
of the line. Let I(t) be the number for which the two neighborhoods
lie on opposite sides of the line, and II(T) be the number
for which the two neighborhoods
lie on the same side. Note that C, F, I(t), and II(t) are all
zero if ZX is a convex curve. Various examples suggest that
C + F/2 + I(t) - II(t) = 0, and this was proven by Halpern [2]
for particular smooth curves.

This article provides an elementary proof of this conjecture
for polygonal planar.curves. This result will generalize to
mappings of 1-manifolds into the plane. The techniques used
will relate somewhat to the polyhedral analogue of catastrophe
theory.