Viktor Blåsjö in The Isoperimetric Problem писал(а):
Steiner gave five proofs of the isoperimetric theorem. Lovely as they are, he left one point open to attack: all proofs assume the existence of a solution (his strategy is always to take a figure that is not a circle and show that its area can be improved). This did not go unpunished. The analyst vultures can smell an existence assumption from miles away. To this day, many authors, revealing their sympathies, are eager to point out that existence is nontrivial. Perron [30] at least jokes about it:
Theorem. Among all curves of a given length, the circle encloses the greatest area.
Proof. For any curve that is not a circle, there is a method (given by Steiner) by which one finds a curve that encloses greater area. Therefore the circle has the greatest area.
Theorem. Among all positive integers, the integer 1 is the largest.
Proof. For any integer that is not 1, there is a method (to take the square) by which one finds a larger positive integer. Therefore 1 is the largest integer.
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27.04.2023, 01:12 
