I should mention here that the problem generalizes to higher finite dimensional spaces. On Fri, Aug 30, 2013 at 2:57 PM, Cris Moore <moore@santafe.edu> wrote:
I agree that the way the lobes of the baseball seam curve in towards each other isn't optimal. So another possibility is a series of semicircles drawn on 4 faces of a cube (you see what I mean, right?)
Cris
On Aug 30, 2013, at 12:11 PM, Andy Latto <andy.latto@pobox.com> wrote:
What's the maximum volume if require the curve to be closed?
In two dimensions, the answer is a circle; the circle encloses the most area, so it is the best convex solution. And if you have a nonconvex closed curve, you can replace it with the boundary of its convex hull, which is always a shorter curve, and scale up.
But I have no idea what the best solution is in 3 dimensions; here maybe a baseball-seam shaped curve is correct. But I remember Conway claiming on this list that every extremal mathematical property he had ever seen claimed for the baseball-seam curve was false, so either there's something better, or Conway was unaware of this problem.
Andy
Andy
On Fri, Aug 30, 2013 at 1:50 PM, Allan Wechsler <acwacw@gmail.com> wrote:
I wonder if they knew of the work of P. A. P. Moran that Erich referred to.
On Fri, Aug 30, 2013 at 1:34 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Apologies, but I misunderstood the problem as saying that the curve has to *lie* on the unit sphere.
The problem Allan asks was a problem that George Bergman and Wolfgang Kahane of U.C. Berkeley worked on informally around 2003 and solved -- and it was pretty hard. But I don't think they published, and I don't know what the answer is.
--Dan
On 2013-08-30, at 9:02 AM, Allan Wechsler wrote:
Which three-dimensional continuous differentiable curve of unit arc-length encloses the greatest volume in its convex hull? _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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