What is the smallest (either in terms of perimeter or area) convex shape in $R^2$ such that all curves of length 1 are properly contained within it ? I'm thinking a right triangle with sides of $1/\sqrt(2)$, but I'm not sure how to reason about this rigorously. Any pointers would be appreciated.
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Do you allow candidate curves to be shifted and rotated? – David G. Stork Aug 13 '16 at 00:09
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1This is an open question called Moser's worm problem. MathOverflow discussion: http://mathoverflow.net/q/32477 – Aug 13 '16 at 00:13