https://www.cut-the-knot.org/do_you_know/isoperimetric.shtml

proved that H => A. That is, for a fixed perimeter, if a shape of maximal area exists (H), then it must be a circle (A). It then discussed some issues regarding the existence hypothesis (H). My questions are:

(1) Why does "a shape with the smallest area for a given perimeter" not exist? How does one prove this?

(2) How does one prove, H, the existence hypothesis? Can someone please outline the "limiting procedure which is quite simple but requires some basic elements of Calculus"?

(3) Can someone further clarify the need to prove H?

Thank you very much.

Just by example: consider a rectangle with one dimension "almost half the given perimeter" and the other "almost zero". The area of such a rectangle is "almost zero" and can be made arbitrarily small while the perimeter remains fixed.

>
>(2) How does one prove, H, the existence hypothesis? Can
>someone please outline the "limiting procedure which is
>quite simple but requires some basic elements of Calculus"?

There are several steps, but essentially one shows that among isoperimetric polygons, the regular ones have the largest area and that the area of a regular polygon is less than that of the isoperimetric circle, with the difference in area tending to 0.

>
>(3) Can someone further clarify the need to prove H?

Not every infinite set contains extremal elements. For example, the set of all fractions less than 1 does not have an extremal element. The isoperimetric problem proper, if restricted to polygons, has no solution.