TITLE: Smallest area shape that covers all unit length curve
QUESTION [16 upvotes]: On a euclidean plane, what is the minimal area shape S, such that for every unit length curve, a translation and a rotation of S can cover the curve.
What are the bounds of the shape's area if this is a open problem?
When I asked this problem few years ago, someone told me it's open. I don't know if this is still open and I can't find any reference on it. 
I don't even know what branch of mathematics it falls under. so I can't even tag this question.

REPLY [10 votes]: The result of mine alluded to in A B's answer, giving an improved lower bound, is now on arxiv: http://arxiv.org/abs/1101.5638