I was thinking of what way to loop a string of a given length such that area inside is maximum. The reason I mentioned smooth in the question is because if we had a mathematical string, then we could deform tiny segments making bumps on the perimeter and reach an infinite perimeter.
My attempt: Let's assume the curve to be $\gamma(t)$ and let the parameter domain be $\left[ 0, t_{max} \right]$ then the arclength is given as:
$$ L = \int_0^{t_{max} } \sqrt{ \dot{\gamma} \cdot \dot{\gamma} } dt $$
Note: The above arclength is fixed.
The area of the parametric curve maybe given by gauss's shoe lace formula in vector form:
$$ A= \int_0^{t_{max} } \gamma \times \dot{\gamma} dt$$
And, now, we want to optimize:
$$ S = \frac{A}{L} = \frac{\int_0^{t_{max} } \gamma \times \dot{\gamma} dt}{\int_0^{t_{max} } \sqrt{\dot \gamma \cdot \dot{\gamma} } dt}$$
I suppose the above is a calculus of variations question, but I haven't learned the tools to solve it. Still, could someone write a solution? I am interested in what would be the answer to this optimization problem.
Turns out this problem well known as dido's problem and the answer is a circle.