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Is it true that the perimeter of any convex polygon in the unit disk on the Euclidean plane is less than the circumference $2\pi$ of the circle? Thanks. [The OP has solved it]

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    Since the upper bound $2\pi$ cannot be attained by the perimeter of a convex polygon, a better word than "maximum" in your title would be "supremum", i.e. least upper bound. (Which is $2\pi$, as you've asked in the body of the question.)2012-12-07

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