What's the smallest area that can contain all curves of length 1?

Question

brittanydosey · Answer

Theorem. A closed plane curve of length L and curvature bounded by K can be contained inside a circle of radius L/4−(π−2)/2K.

kainui · Answer

Honestly I have no idea what the answer is, anyone have any ideas? Maybe like some upper or lower bound to get started?

brittanydosey · Answer

would that help?

kainui · Answer

It might, since a subset of all curves of length 1 is all the closed curves of length 1.

brittanydosey · Answer

yes

kainui · Answer

If we think about every concave shape we can create a larger area convex shape by reflecting past the indented parts |dw:1428499082724:dw| Also since I see no reason to restrict it from curves that wrap on themselves or maybe just slightly off, we can sorta fill the entire interior of that closed shape with other lines of length 1 like this: |dw:1428499177395:dw| where the wrappings are an infinitesimal distance apart so that it's not completely on top of itself.... Which might simplify the problem a bit.