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Problem 62: Volume Maximizing Convex Shape
 Statement
 Let C be a convex piece of paper; its
boundary may be a smooth curve, or a polygon.
A perimeter halving folding is a folding of C obtained by
identifying two points x and y on the boundary of C that halve
the perimeter, and then folding C by ``gluing'' xy to yx.
This always results in a unique convex shape in 3D, a polyhedron if
C is a convex polygon [DO07b].
What unitarea shape C achieves the maximum volume
possible via a perimeterhalving
folding?
 Origin
 Posed by
Joseph Malkevitch in 2002, in a slightly different form:
for polygons, and not restricting the folding to perimeterhalving.
The modifications above were suggested at CCCG'05 [DO06].
The restriction to perimeter halving eliminates some more complex foldings
possible for some convex polygons, and so in that sense simplifies
the problem.
The extension to smooth shapes is a natural generalization.
Smooth shapes only admit perimeterhalving foldings.
 Status/Conjectures
 Open.
 Partial and Related Results
 Even fixing the shape and finding the maximum volume perimeter
halving for that shape is difficult.
For a circular disk, all perimeter halvings lead to a flat doublycovered
half disk, all of volume zero.
The only other shape for which
the answer is known, and then only empirically,
is the case of C a square [ADO03].
The resulting polyhedron of 6 vertices and 8 faces, shown in
Fig. 3,
achieves about 60% of the volume of
a unitarea sphere.
Figure 3:
The maximum volume convex polyhedron foldable from a square.

 Appearances
 [DO06].
 Categories
 folding and unfolding
 Entry Revision History
 J. O'Rourke, 26 Aug. 2005.
 ADO03

Rebecca Alexander, Heather Dyson, and Joseph O'Rourke.
The convex polyhedra foldable from a square.
In Proc. 2002 Japan Conf. Discrete Comput. Geom., volume 2866
of Lecture Notes Comput. Sci., pages 3850. SpringerVerlag, 2003.
 DO07b

Erik D. Demaine and Joseph O'Rourke.
Geometric Folding Algorithms: Linkages, Origami, Polyhedra.
Cambridge University Press, 2007.
In press. http://www.gfalop.org (formerly
http://www.fucg.org).
 DO06

Erik D. Demaine and Joseph O'Rourke.
Open problems from CCCG 2005.
In Proc. 18th Canad. Conf. Comput. Geom., pages 7580, 2006.
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The Open Problems Project  September 19, 2017