G-2006-72
Isoperimetric Polygons of Maximal Width
Charles Audet, Pierre Hansen, and Frédéric Messine
The value
is shown to be an upper bound on the width of any n-sided polygon
with unit perimeter.
This bound is reached when n is not
a power of 2,
and the corresponding optimal solutions are the
regular polygons when n is odd,
and clipped regular Reuleaux polygons when n is even but not a power of 2.
Using a global optimization algorithm, we solve the problem for
n =4. The optimal width for the quadrilateral is shown to be
We propose two mathematical programs to determine the maximal width
when n =2s with
and provide approximate, but
near-optimal, solutions obtained by various heuristics and local
optimization for n =8,16 and 32.
Published November 2006 , 24 pages