I'm not sure if this curve is supposed to be a closed curve, or whether it's allowed to cross itself. But I don't think the answer would be any different in those 4 cases. What would change the answer is if the curve need not be the image of a closed interval, but instead can be the image of a half-closed or open interval. If the curve is the image of a closed interval, then you can get convex hulls arbitrarily close to the volume of the unit ball = 4pi/3, but necessarily less than that. But if the curve is instead the image of a half-closed or open interval, then you can reach 4pi/3 by arranging that the curve be dense in the unit sphere, which isn't hard to do, so I'll leave that as an exercise for the reader. --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