And just to add to the profusion of problems, what if the curve -- of length L -- is constrained to lie on the unit sphere? Let the volume of the maximum convex hull of such curves be denoted by f(L). Then what is the function f ??? And of course the same thing could be asked for *closed* curves of length L constrained to lie on the unit sphere. That maximum could be denoted by g(L). What's g ??? We are *assuming* that the sup of all the volumes in either case is actually attained, since intuitively that seems likely, and probably not too hard to prove. But just in case, we should probably redefine f(L) (and g(L)) to be the sup of volumes of convex hulls over all curves (resp. closed curves) of length L on S^2. --Dan On 2013-08-30, at 12:36 PM, Dan Asimov wrote:
Not to mention infinite-dimensional Hilbert space.
--Dan
On 2013-08-30, at 12:23 PM, Allan Wechsler wrote:
I should mention here that the problem generalizes to higher finite dimensional spaces.
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