minimize ratio perimeter/sqrt(area​) of a random shape

Random search used. Result is circle of cause.

http://simulations.narod.ru/
Shape is represented as 32-vertex polygone. The program in starts with random shape. In iteration of random search it search minimum perimeter/sqrt(area) each time. In each iteration the code try random position of vertices near current positions. Zone of random search is contracted exponentially. Contraction ratio is 0.99. 1000 iterations are used. So it is 0.99^1000=0.000043 final precision. the shape size is normalized to sqrt(area) each time. Only circle has minimal ratio perimeter/sqrt(area) so as result we have circle.

Run minimal_perimeter.m to see animation.
Function perimeter.m calculates perimeter of the polygone.

minimal_perimeter_avi.m is to make avi-file
minimal_perimeter_gif.m is to make gif-file.