Shape with the Best Area to Perimeter Ratio

Finding the shape that gives the greatest area for a fixed perimeter
Time stamps:
0:00 Introduction
1:11 Concave Shapes
2:02 Convex Shapes
3:18 Maximising triangular area
4:32 Answer

[elyades]

nice concept and funny video, had a good time !

[agrajyadav]

This is a very clever proof!
I have a question - can we argue that since we can split the shape into half and maximise that, why not a fourth, an eight, a 2^nth, and as n grows to infinity, the shape has to be perfectly rotationally symmetric, which is a circle. Are there flaws in this reasoning?