Suppose you are given a single unit square, and you are permitted to cut it into $k$ (connected)
pieces (where $k=1$ means just the square). Your task is to construct the largest volume
convex body in $\mathbb{R}^3$ by pasting the $k$ pieces together as its complete surface;
so its surface area is $\le 1$.
Denote its volume by $V_\max(k)$.
(This is a more general version of my earlier question, [Covering a Cube with a Square][1].)
The best one could hope for is to create a sphere with unit surface area,
when the radius satisfies $4 \pi r^2 =1$ and so $r=\frac{1}{2\sqrt{\pi}} \approx 0.28$,
and the volume is
$V_\max(\infty)= V_\max = \frac{4}{3} \pi r^3 = \frac{1}{6 \sqrt{\pi}} \approx 0.094$.
In response to a question of Joe Malkevitch, with students I computed the optimum for $k=1$,
with no cuts and no overlap,
and found a maximum volume of
$V_\max(1)\approx 0.056$, which is about 60% of the sphere volume.
[p.418 in *[Geometric Folding Algorithms: Linkages, Origami, Polyhedra][2]*,
Demaine, O'Rourke,
Cambridge, 2007.]:
<br /> ![Max Volume Polyhedron][3]<br />
The seeming randomness of this shape discouraged my further pursuit.
The case $k=2$ resembles the famous [tea-bag problem][4], but here I am restricting attention to convex bodies.
As $k \to \infty$, one should be able to approach $V_\max$ by, for example, cutting the square into many nearly equilateral triangles and constructing a [geodesic dome][5].
I am wondering if anyone has heard of this problem in any guise before?
And in any case, can anyone see any clear hypothesis, for any particular $k$?
Presumably one could establish that $V_\max(k+1) > V_\max(k)$, but beyond that,
I do not see how to make inroads.
Is there an obvious candidate for $k=2$?
<b>Update</b>. Here is my interpretation of Gerhard's suggestion for $k=2$:
<br /> ![Square to Box][6]
[1]: http://mathoverflow.net/questions/95867/
[2]: http://gfalop.org/
[3]: http://cs.smith.edu/~orourke/MathOverflow/MaxVol.jpg
[4]: http://en.wikipedia.org/wiki/Paper_bag_problem
[5]: http://en.wikipedia.org/wiki/Geodesic_dome
[6]: http://cs.smith.edu/~orourke/MathOverflow/SquareBox221.jpg