Consider a rope of fixed length $ k $. If we link the extremes of the rope in order to form a closed curve, what is the maximun area that we can cover?
More formally:
Let $ k \in \mathbb{R}^+ $. We consider all piecewise $ C^1 $ closed curves $ \delta : [a,b] \subset \mathbb{R} \longrightarrow \mathbb{R}^2 $ such that $ L_a^b ( \delta ) = k $ (length of $ \delta $). For each $ \delta $, let be $ D_{ \delta } \subset \mathbb{R}^2 $ such that $ \partial D_{ \delta } = \delta([a,b]) $ (the surface cover by $ \delta $). What is the maximun value of $$ \int_{ D_{ \delta } } 1 $$ over all the possible curves $ \delta $?
I tried to use Green's Theorem because it gives a relation between the curve $ \partial D_{ \delta } $ and the area of $ D_{ \delta } $. If $ \delta(t) = ( \delta_1 (t), \delta_2 (t) ) $ and $ F(x,y) = (0,x) $, then Green's Theorem allow us to write $$ \int_{ D_{ \delta } } 1 = \int_{ \partial D_{ \delta } } F = \int_a^b F( \delta_1 (t), \delta_2 (t) ) \cdot (\delta_1' (t), \delta_2' (t) ) \ dt = \int_a^b \delta_1 (t) \cdot \delta_2' (t) \ dt $$ while $ L_a^b( \delta ) = \int_a^b \sqrt{ \delta_1' (t)^2 + \delta_2' (t)^2} \ dt $.
It gives an expresion of that we want to maximize and the restriction, and both expresion are similar in a certain way, but this does not seem useful (at least for me). Intuitively, I think that the solution is to consider a circle of lenght $ k $, but I don't know how to prove it.
