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Mosers worm problem is the problem of finding a shape of smallest area that can cover all curves of unit length. The shape is allowed to be rotated and translated to cover the curve. (Make a blanket of minimal size that can cover a worm of length 1dm). I work with a version of the problem where the shape is required to be convex.

I wonder if anyone knows a result of the following type:

There is a smallest shape W where W contains the points (0,0) and (0,1) and W is contained in the strip $0\le y \le 1$.

I thought this would be obvious but have failed to prove it and have no idea how to. This would help me a lot with computer search for better lower bounds.

Does anyone know of somethings like this? (Or can come up with a proof?)

Thanks

David