If you were able to join the end such that you didn't change the length of the string, you could create a circle. If instead of creating a circle with a given area, you instead pulled on both sides, collapsing the enclosed area to zero. You would have something with the perimeter of L still, but now no area enclosed (basically two parallel pieces of string with length L/2 touching each other). Area is two dimensional, perimeter is a length. As has been noted, maximizing enclosed space to perimeter results in a circle. Minimizing enclosed space to perimeter results in a shape where one side of the shape touches the other, most easily thought of as collapsing to a single line.