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What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?

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If the optimal object is not a round sphere then the answer is not known. (There are few exceptions, but it is almost true.) –  Anton Petrunin Dec 9 '11 at 4:14
It is definitely not a sphere. –  Vladimir Reshetnikov Dec 9 '11 at 4:22
I nominate a fractional orange slice: A slightly warped semicircular arc and part of a second one in a plane perhaps 120 degrees from the first one, sharing common end points. I leave the details to those with the computational power. Gerhard "Not Ready To Compute Volumes" Paseman, 2011.12.08 –  Gerhard Paseman Dec 9 '11 at 5:42
related: mathoverflow.net/questions/26212 –  Steve Huntsman Dec 9 '11 at 22:26
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up vote 18 down vote accepted

I believe this problem has been mentioned a few times in the literature, and has been solved for certain restrictions on the curve. For example if the curve has no four coplanar points then the maximal volume is achieved by one turn of a circular helix of height $\frac{1}{\sqrt{3}}$ and base radius $\frac{1}{\pi\sqrt{6}}$, this is due to Egervary. For more references see section A28 of "Unsolved problems in geometry" by H.T. Croft, K.J. Falconer, R.K. Guy. Melzak and Schoenberg have treated the corresponding problem for closed loops (Schoenberg has treated even dimensions), and have given answers under similar restrictions. In no case is a complete answer known.

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