Volume-maximizing convex shape under perimeter-halving folding
Let $C$ be a convex region of the plane, with smooth or polygonal boundary. A perimeter-halving folding of $C$ chooses boundary points $x$ and $y$ that split the perimeter into two arcs of equal length and glues the two arcs together, each point of one arc to the point of the other at the same arc length from $x$. The result is always a unique convex body in $\mathbb{R}^3$, a polyhedron when $C$ is a polygon.
Already for a fixed shape the largest volume is hard to find. For a disk every perimeter halving gives a flat doubly covered half-disk, of volume zero. The only other known case is the square, and only empirically (Alexander, Dyson and O'Rourke): the best folding is a polyhedron with $6$ vertices and $8$ faces, of about $60\%$ of the volume of a sphere with the same surface area.
Problem. Which convex shape $C$ of unit area attains the largest volume obtainable from it by a perimeter-halving folding?
Source: Malkevitch (2002), for polygons and arbitrary foldings; recast at CCCG 2005 (Demaine and O'Rourke, 2006); Problem 62 of The Open Problems Project (Demaine, Mitchell and O'Rourke).
Imported from topp.openproblem.net.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- OC
- Keywords
- Alexandrov gluing, folding convex polygons, perimeter halving, The Open Problems Project
References
@book{demaine2007gfalop,
author = {Demaine, Erik D. and O'Rourke, Joseph},
title = {Geometric Folding Algorithms: Linkages, Origami, Polyhedra},
publisher = {Cambridge University Press},
year = {2007},
url = {https://gfalop.org/},
}
@inproceedings{demaine2006cccg05,
author = {Demaine, Erik D. and O'Rourke, Joseph},
title = {Open problems from {CCCG} 2005},
booktitle = {Proc. 18th Canadian Conference on Computational Geometry},
year = {2006},
pages = {75--80},
}
@inproceedings{alexander2003,
author = {Alexander, Rebecca and Dyson, Heather and O'Rourke, Joseph},
title = {The convex polyhedra foldable from a square},
booktitle = {Proc. 2002 Japan Conference on Discrete and Computational Geometry},
series = {Lecture Notes in Computer Science},
volume = {2866},
publisher = {Springer-Verlag},
year = {2003},
pages = {38--50},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01100, "Volume-maximizing convex shape under perimeter-halving folding", clopen.solutions.