A closed curve with a certain perimeter must be convex if the closed area is the largest.

If the figure is concave, you can use the concave part of the curve to make a straight line with two intersections, and make a symmetrical curve with an arc between the two intersections, so that you can get a figure with the same circumference and a larger area.

2

In the closed curve with the largest area, if point A and point B divide its circumference equally, then the chord AB divides its area equally.

If AB doesN't divide its area equally, then the figure must have a larger area on one side of AB, as shown in the figure, so let's set N>m, and then remove the symmetrical figure n' about AB with m as n, and the figure composed of N and N' is the same as the original perimeter, but the area is larger.

For chord AB with equal perimeter and area, only half of the graphs on both sides of AB are considered. If c is any point of this arc, then ∠ ACB = 90. Otherwise, the graph can be divided into m, n and p, and only the size of ∠ ACB needs to be changed to make ∠ ACB = 90, then m

This shows that this half of the curve must be a semicircle, so the other half is also a semicircle.

Of course, this proof assumes that there is a maximum.

If this assumption is removed,

Elementary mathematics is powerless. . .