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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 unit-area shape C achieves the maximum volume possible via a perimeter-halving folding?
Origin
Posed by Joseph Malkevitch in 2002, in a slightly different form: for polygons, and not restricting the folding to perimeter-halving. 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 perimeter-halving 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 doubly-covered 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 unit-area sphere.
Appearances
[DO06].
Categories
folding and unfolding
Entry Revision History
J. O'Rourke, 26 Aug. 2005.

## Bibliography

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 38-50. Springer-Verlag, 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 75-80, 2006.

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The Open Problems Project - September 19, 2017