Isoperimetric Theorem and Inequality

Isoperimetric theorem has been known from the time of antiquity. Elegant all by itself, the theorem also has a fascinating history. The most important contribution towards its rigorous proof was given in 1841 and is due to Jacob Steiner (1796 - 1863). At that time, it stood as the center of controversy between adherents and analytic (i.e. using Calculus) and synthetic (pure Geometric) methods. Although accepting the validity of the analytic methods, Steiner only used the synthetic approach. His proof contained a flaw that was later fixed by using the analytic apporoach. To better understand the nature of this discussion, let me first formulate 2 equivalent statements.

Statement 1

Among all the planar shapes with the same perimeter, the circle has the largest area.

Statement 2

Among all the planar shapes with the same area, the circle has the shortest perimeter.

Theorem

Statements 1 & 2 are the same.

Proof

As a shorthand, denote Statement 1 as A and Statement 2 as B. Assume A holds and let us prove B. Supposed, on the contrary, that B is false. Then for a given circle C there exists a figure F with the same area but with a perimeter shorter than C. Shrink C into a smaller circle C' whose perimeter equals that of F. The area of C' will clearly be smaller than that of C and consequently, it will be smaller than the area of F. Now, this contradicts our assumption that A holds: C' and F have the same perimeter but the area of the circle C' is less than the area of F. Thus A=> B.

The implication B=>A is proven in precisely the same fashion.

Note that neither A or B was proven but rather, that they are either both true or both false. Mathematical theorems in general have premises and conclusions and assert that the latter follow from the former. Human reasoning often resorts to the well known modus pones syllogism: if A=>B and A is true, then so is B. Oftentimes, the premise A seems to be so obviously true that a proof of A=>B is mistakenly substituted for a proof of B.