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.



Smallest area shape that covers all unit length curve
Whereas I don't know of any recent progress in this problem, let me mention one result for 
*closed* curves.



> 
> **Theorem.** A closed plane curve of length $L$ and curvature bounded by $K$ can be contained inside a circle of radius $L/4 - (\pi - 2)/2K$. 
> 
> 
> 


This was proved in 1974 by H.H. Johnson ([link 1](http://www.ams.org/mathscinet-getitem?mr=0348631)) who used calculus of variations methods. A geometric proof was given a bit later by Chakerian, Johnson and Vogt ([link 2](http://www.ams.org/journals/proc/1976-057-01/S0002-9939-1976-0402611-2/home.html%20)).




---


**Edit.** Apparently the problem is still open. Here's an article ([arXiv link](http://arxiv.org/abs/math.MG/0701391)), which contains a survey of some known results as of 2009. From the Introduction:



> 
> In 1966, Leo Moser asked for the region of smallest area which can accommodate
>  every planar arc of length one. The problem is known as “Moser’s worm problem” and is a variation of universal cover problems. In Moser’s problem, a cover is a set which contains a copy of any rectifiable planar arc of unit length, and is usually assumed to be convex. Such a minimal cover is known to have area between 0.2194 and 0.2738. However, the
>  original problem remains unsolved.
> 
> 
> 


