I have a partial answer:

You can always assume that is an arc length parametrization ( for all )

You have , so that , where are the unit tangent, normal vectors at time (the normal is assumed to be the image of by the rotation of angle ) and is the (algebraic) curvature at time . As a consequence, the length of the curve is: since the convexity of the curve gives (I assume in fact that the curve is oriented positively, so that the result you give is right; otherwise we could not check whether ).

So, if you know that the "global curvature" for a convex curve equals , you are done. This is a classical result, and I have a two page long proof in a book I don't feel like rewriting here. So I hope you know that...